Classical Analysis, Operator Theory, Geometry of Banach Spaces ...

Last Periodic report- Part A - Research ResultsA1 . Scientific HighlightsTask 1 (Hardy and Bergman spaces,interpolation)Following the paper [3] already mentioned in the previous report, Nicolau and Dyakonov studied in [2] thesequences of elements of the unit disc such that any sequence of bounded nonvanishing values can be interpolatedby a bounded analytic function without zeroes. Free interpolating sequences in the Nevanlinna and Smirnovclasses are discussed in [4], in which some partial results are also obtained for the problem of describing thosefunctions which admit an harmonic majorant.During his visit to Barcelona, the postdoc X.Dussau obtained ”weighted Polya-Plancherel inequalities” whichrelate the behaviour of some entire functions on the real line to their behaviour on the integers. Also a joint paperby F.Perez-Gonzales (La Laguna, team 3) and the postdoc J.Rattya provide new sampling results for a class ofspaces related to Hardy and Bergman spaces. There is also a joint work in progress by A. Borichev (team 1) andY.Lyubarskii (team 7) concerning maximal decay of nonzero functions along appropriate uniqueness sequencesin various spaces of analytic functions.Related with sampling and interpolation, this time for Hardy and Bergman spaces in the disc, anothersignificant contribution is given in [5]. Here the uniform densities describing exactly interpolating sequences forthe Bergman spaces are related to harmonic measure of certain ”cheese domains” obtained by deleting suitablehyperbolic disks. It seems to be the first time that this central notion in complex analysis is brought into thepicture of interpolation and sampling.Task 2 (Cauchy integrals , Capacities , Harmonic approximation)The paper [2], by J.Pau (team 3) and S. J. Gardiner (team 4) dealt with the representation, or approximation,of functions on the boundary of a domain Ω ⊂ R n by sums of Poisson, Green or Martin kernels associated witha set E ⊂ Ω, and with the related issue of whether E can be used to determine the suprema of certain harmonicfunctions on Ω. The paper ”Smooth potentials and harmonic estimates” by A.Gustafsson (postdoc in team 3)and S.J.Gardiner (Publ. Mat. 48 (2004), 241-249) examines when it is possible to find a smooth potential ona C 1 domain D with prescribed normal derivatives at the boundary. It is shown that this is always possiblewhen D is a Liapunov-Dini domain, and this restriction on D is essential. An application concerning C 1superharmonic extension is given in response to a problem posed by Verdera, Melnikov (both of Team 3 of theNetwork) and Paramonov.The central role of the Cauchy kernel within Calderon-Zygmund theory in two dimensions is clarified in thepreprints ”L 2 boundedness of the Cauchy transform implies L 2 boundedness of all Calderon-Zygmund operatorsassociated to odd kernels”, by X.Tolsa, and ”Estimates for the Cauchy integral over Ahlfors regular curves”, byX.Tolsa and M.Melnikov.Task 3 (Function models and applications of operator theory)The study of the generalized Schur algorithm for Nevanlinna functions (the line case) was continued andimportant new results about the factorizations of J–unitary matrix polynomials, of operator representationsof generalized Nevanlinna functions and their applications in singular rank one perturbations of self-adjointoperators were obtained, see the joint papers [1], [2], [6], [7]. These results are also related via extensionproblems for symmetric operators to Nevanlinna-Pick interpolation problems and as such to the results of”Solution of a multiple Nevanlinna-Pick problem” by H.Langer (team 8) and A.Lasarow (J. Math. An. andAppl. 293 (2004), 605-632), where a description of the solutions of a Nevanlinna-Pick problem by means oforthogonal functions was obtained.3

In the joint paper [12 ], A.Hartmann (team 1), D.Sarason and K.Seip (team 7) gave a characterization ofthose surjective Toeplitz operators on the Hardy spaces which have a nontrivial kernel in terms of the extremalfunction of the kernel. S.Pott (team 5) and E.Strouse (team 1) found in [14] a necessary and a sufficientcondition for the product of two Toeplitz operators on a weighted Bergman space to be bounded involving theBerezin transform of the symbols of these Toeplitz operators, and N.Nikolski (team 1) studied the link betweencondition numbers (a notion useful in numerical analysis) and spectral condition numbers for Toeplitz operatorson l p (Z + ), 1 ≤ p ≤ +∞ (these phenomena involve a critical number k(p), equal to 2 for p = 1 and p = +∞,equal to one for p = 2, of value unknown for 1 < p < +∞, p ≠ 2). In the joint paper [10], I. Gohberg (team 9),M.A. Kaashoek (team 2) and F.van Schagen (team 2) found new inversion formulas for finite Toeplitz operators.Task 4 (The invariant subspace problem)In the joint paper [3], I.Chalendar (team 1), J.R.Partington (team 5) and R.Smith (team 5) show that theexistence of pairs (x, y) of elements of H such that < T n x, y >= ˆf(−n) ( n ≥ 0) for some specific f ∈ L 1 (T) doesimply the existence of nontrivial invariant subspaces for T. In the same paper they also establish for the firsttime a link between the Brown approximation scheme and the Hilbert space version of the Atzmon-Godefroymoment theorem.A.Atzmon (team 9) and B.Brive (team 1, postdoc at Tel-Aviv) characterized in [1] surjectivity of lineardifferential operators of infinite order with constant coefficients on weighted Hilbert spaces of entire functionswith radial log-convex weight. Sophie Grivaux, a young Mathematician from the Paris team, proved in particularthat each operator on the separable Hilbert space can be written as the sum of two hypercyclic operators.In the joint papers [6] [8] S.Grivaux and F. Bayart point out the role of the unimodular point spectrum inhypercyclicity theory, using sophisticated measure theoretical tools, and they develop in [7] the natural strongernotion of frequent hypercyclicity, studying various concrete examples.Task 5 (Geometry of Banach Spaces and applications)G. Pisier (team 6) proved that the only quotients of the direct sum of the row and column operator spacesthat embed in a semifinite non-commutative L 1 space are R, C, R + C (the classical Gaussian case) and 4 otherspaces built out of these first 3. Also Q.Xu (team 6) developed complex interpolation of certain row and columnoperator spaces.In St Petersburg (team 10) S.Kysliakov and his graduate student D.Anisimov established Interpolation andcorrection results for some function spaces defined in terms of certain ”double singular integral operators”. Thisapplies to many classical operators of harmonic analysis (for instance, the Hardy-Littlewood square function).Task 6 (Convex geometry,concentration of measures)D. Cordero, M. Fradelizi and B. Maurey ( team 6) verified an inequality of multiplicative concavity conjecturedby Banaszczyk relative to the gaussian measures of dilations of any given centrally symmetric convexset (the so-called “(B)-conjecture”). G. Aubrun and M. Fradelizi, also from team 6, proved a conjecture ofSchneider. O. Guédon (team 6) and G. Paouris (a young researcher appointed by the Paris-6 node) studied in ajoint paper the concentration of mass of the unit ball of the finite dimensional Schatten class S n p (with respectto the Hilbert-Schmidt norm).After solving the duality problem for metric entropy in the central euclidean case in a paper mentioned inthe last report V. Milman (team 9), A. Artstein (former predoc at team 6 and graduate student of team 9)and S.Szarek (team 6) introduced with N.Tomczak-Jaegermann the notion of ”convexified separation”. Preciseduality results for packing and ”convex packing” numbers are obtained for K-normed spaces, and this notionof convex separability should play in the future an important role in complexity theory and optimization.4

A2.Joint publications and patentsYoung Researchers are indicated using bold. We indicate here the joint papers published and the jointpreprints completed by coauthors from different nodes during the final year of network activity.period coveredby the report, the joint preprints which were completed during that period. We included in the list two jointpapers by C. Mehl (postdoc at Amsterdam for two months) and one member of the Amsterdam team (togetherwith an external expert), and a joint paper by H.Langer (coordinator of team 8) and A.Lasarow, who waspostdoc in team 8.Task 1 (Hardy and Bergman spaces, interpolation)1. A.Borichev (team 1), F.Nazarov and M.Sodin (team 9), Lower bounds for quasianalytic functions, Math.Scand., to appear.2. K.M. Dyakonov (team 10) and A.Nicolau (team 3), Free interpolation by nonvanishing analytic functions,preprint.3. A.Nicolau (team 3), J.Ortega-Cerda (team 3) and K.Seip (team 7) The constant of interpolation, Pac. J.Math. 213 (2004), 389-398.4. J. Ortega-Cerda (team 3) and K. Seip (team 7), Harmonic measure and uniform densities, Indiana J.Math., to appear.5. A. Hartmann (team 1), X.Massaneda (team 3), A. Nicolau (team 3) and P.Thomas Free Interpolation inthe Nevanlinna and Smirnov classes and harmonic majorants, J. Func. An., to appear.Task 2 (Cauchy integrals , Capacities , Harmonic approximation)1. S.Gardiner(team 4) and J.Pau (team 3), Approximation in the boundary and sets of determination forharmonic functions, Illinois J. Math. 47 (2003), 1115-1136.Task 3 (Function models and applications of operator theory)1. D. Alpay, T.Ya. Azizov, A. Dijksma (team 2), and H. Langer (team 8), The Schur algorithm for generalizedSchur functions III: Factorizations of J-unitary matrix polynomials, Lin. Alg. Appl. 369 (2003), 113–144.5

2. D. Alpay, A. Dijksma (team 2), and H. Langer (team 8),Factorization of J-unitary matrix polynomials onthe line and a Schur algorithm for generalized Nevanlinna functions, Lin. Alg. Appl. 387, 2004, 313-342.3. D. Alpay, T.Ya. Azizov, A. Dijksma (team 2), H. Langer (team 8), and G. Wanjala (team 2), A basicinterpolation problem for generalized Schur functions and coisometric realizations, Operator Theory: Adv.,Appl., 143, Birkhäuser Verlag, Basel, 2003, 39-76.4. D. Alpay, T.Ya. Azizov, A. Dijksma (team 2), H. Langer (team 8), and G. Wanjala (team 2), The Schuralgorithm for generalized Schur functions IV: unitary realizations, Operator Theory: Adv., Appl., 149,Birkhäuser Verlag, Basel, 2004, 23-45.5. A. Batkai, P. Binding, A. Dijksma (team 2), R. Hryniv, and H. Langer (team 8), Spectral problems foroperator matrices, preprint.6. A. Dijksma (team 2), H. Langer (team 8), and Y. Shondin, Rank one perturbations at infinite coupling inPontryagin spaces, J. Funct. Anal. 209 (2004), 206–246.7. A. Dijksma (team 2), H. Langer (team 8), A. Luger (team 8), and Y. Shondin, Minimal realizations ofscalar generalized Nevanlinna functions related to their basic factorization, Operator Theory: Adv., Appl.,to appear.8. I. Gohberg (team 9), S. Goldberg and M.A. Kaashoek (team 2), Basic Classes of Linear Operators,Birkhäuser Verlag, Basel, 2003; 423 pp.9. I. Gohberg (team 9), M.A. Kaashoek (team 2) and F. van Schagen (team 2), On inversion of convolutionintegral operators on a finite interval, in: Operator Theoretical Methods and Applications to MathematicalPhysics. The Erhard Meister Memorial Volume, OT 147, Birkhäuser Verlag, Basel, 2004, pp. 277–285.10. I. Gohberg (team 9), M.A. Kaashoek (team 2) and F. van Schagen (team 2), On inversion of finite Toeplitzmatrices with elements in an algebraic ring, Lin.Alg.Appl. 385 (2004), 381-389.11. I. Gohberg (team 9), M.A. Kaashoek (team 2) and A.L. Sakhnovich, Taylor coefficients of a pseudoexponentialpotential and the reflection coefficient of the corresponding canonical system, Math. Nachr.,to appear.12. A. Hartmann (team 1), D. Sarason, and K. Seip (team 7), Surjective Toeplitz operators, available athttp://www.math.ntnu.no/ seip/surjectivetoeplitz.pdf13. B. Jacob, J.R. Partington (team 5) and S. Pott (team 1), Conditions for admissibility of observationoperators and boundedness of Hankel operators, Int. Eq. and Op. Theory 47 (2003), 315-338.14. S.Pott (team 5) and E.Strouse (team 1), Products of Toeplitz operators on the Bergman spaces A 2 α,preprint.15. C.Mehl (postdoc in team 2), A.C.M. Ran (team 2), and L. Rodman, Polar decompositions of normaloperators in indefinite inner product spaces, submitted for publication in Proceedings of 3d Workshop onIndefinite Inner Products, OT-series.6

16. C.Mehl (postdoc in team 2), A.C.M. Ran (team 2), and L. Rodman, Hyponormal matrices and semidefiniteinvariant subspaces in indefinite inner products, submitted for publication in Electronic Journal ofLinear Algebra.17. H.Langer (team 8) and A.Lasarow (postdoc in team 8), Solution of a multiple Nevanlinna-Pick problemvia orthogonal rational functions, J. of Mathematical Analysis and Applications 293 (2004), 605–632.Task 4 (The invariant subspace problem)1. A.Atzmon (team 9) and B.Brive (team 1) Surjectivity and invariant subspaces of differential operatorson weighted Bergman spaces of entire functions, preprint.2. I. Chalendar (team 6), J. R. Partington (team 5) and M. Smith (team 5), Approximation in reflexiveBanach spaces and applications to the invariant subspace problem, Proc. Amer. Math.Soc. 132, (2003),1133-1142.3. I. Chalendar (team 6), J.R. Partington (team 5) and R. Smith (team 5), L 1 factorizations, momentproblems and invariant subspaces, preprint, April 2004.4. I. Chalendar (team 6), L. Habsieger, J.R. Partington (team 5) and T.J. Ransford, Approximate Carlemantheorems and a Denjoy-Carleman maximum principle, Archiv der Mathematik, 83 (2004), 88-96.5. I. Chalendar (team 6) and J.R. Partington (team 5), Convergence properties of minimal vectors fornormal operators and weighted shifts, Proc. Amer. Math. Soc., to appear.6. F. Bayart (team 1), S. Grivaux (team 6), Hypercyclicity: the role of the unimodular point spectrum, C.R. Acad. Sci. Paris, 338 (2004), 703 - 708.7. F. Bayart (team 1), S. Grivaux (team 6), Frequently hypercyclic operators, preprint 2004.8. F. Bayart (team 1), S. Grivaux (team 6), The role of the unimodular point spectrum in hypercyclicity,preprint 2004.Task 5(Geometry of Banach Spaces and applications) and Task 6 (Convex geometry,concentration of measures)1. V.Milman (team 9) and A.Pajor (team 6), Essential uniqueness of M-ellipsoids of a given convex body, toappear in GAFA.2. S.Artstein (team 9), V. Milman (team 9), S. Szarek (team 6) and N. Tomczak-Jaegermann, On convexifiedpacking and metric entropy, to appear in Geom. Funct. Anal.7

FINAL REPORT -Part A- Research resultsA1 . Scientific HighlightsDuring the 48 months of network activity the network made significant progress with respect to all tasks(some truly outstanding results have been obtained by team 3, which resolved long standing questions relatedto task 2). Here below these various achievements are described in some detail.Task 1-Hardy and Bergman spaces, interpolationJ.Ortega-Cerda and K.Seip (teams 3 and 7) obtained in Annals of Maths 155 (2002), 789-806 a descriptionof the Fourier frames of Duffin and Schaeffer, which correspond to the sampling sequences for the classicalPaley-Wiener space of entire functions. Their work is based on de Branges’ theory of Hilbert spaces of entirefunctions. Also X.Massaneda (team 1), J.Ortega-Cerda (team 1) and M.Ounaies were able to describe completelythe interpolating sequences for the Paley-Wiener-Schwartz space of entire functions corresponding tocompactly supported distributions, solving an open problem posed by Ehrenpreis (this result has interesting applicationsto deconvolution equations). Another significant contribution of these authors related with samplingand interpolation is given in Harmonic measure and uniform densities, to appear in Indiana J. Math. Here theuniform densities describing exactly interpolating sequences for the Bergman spaces are related to harmonicmeasure of certain ”cheese domains” obtained by deleting suitable hyperbolic disks. It seems to be the firsttime that this central notion in complex analysis is brought into the picture of interpolation and sampling.A.Hartmann (team 1) and K.Seip (team 7) studied in J. Func. Anal. 202 (2003), 342-262 extremal functionsas divisors for kernel of Toeplitz operators on the Hardy spaces H p , 1 < p < +∞, with applications to thecharacterization of certain complete interpolating sequences (Milestone 2, task 1). They also obtained withD.Sarason a characterization of surjective non-injective Toeplitz operators in terms of the extremal function ofthe kernel, a result also related to the research objectives of task 3.In team 7 Hedenmalm, Jakobsson, and Shimorin obtained a Hadamard type biharmonic maximum principle.Aleman, Beliaev, and Hedenmalm obtained a characterization of the real zero preserving operators that commutewith differentiation on the space of rational functions with a single pole on the real line (J. An., to appear).Lyubarskii solved a non-local interpolation problem for generalized Paley-Wiener space by reconstructing thefunction from the values of its divided differences at points of a complete interpolating sequence.In Pac.J.Math.213 (2004),389-398, A.Nicolau (team 3), J.Ortega-Cerda (team 3) and K.Seip (team 7)manage to relate the constant of interpolation of H ∞ -interpolating sequences with geometrical parameters ofthe sequence, such as the separation constant and others. The paper includes an improved version of theconstructive solution given by P.Jones, giving the optimal bound. Also in the joint preprint Free Interpolationin the Nevanlinna and Smirnov classes and harmonic majorants, to appear in J. Func. Anal., A.Hartmann(team 1), X.Massaneda (team 3), A. Nicolau (team 3) and P.Thomas have studied free interpolating sequencesin the Nevanlinna class in the unit disc and the relation of this problem with the one of describing whichfunctions admit an harmonic majorant. A.Nicolau and K.Dyakonov (teams 3 and 10) studied in the preprintFree interpolation by nonvanishing analytic functions the sequences of elements of the unit disc such that anysequence of bounded nonvanishing values can be interpolated by a zero-free bounded analytic function. This isclosely related to the Pick-Nevanlinna interpolation and the answer is given by the notion of thin sequence.A.Borichev (team 1), A.Volberg (then in team 6) and H.Hedenmalm (team 9) constructed in J. Func. An.207 (2004) 111-160 non z-cyclic functions without zeroes in the disc for a very large class of weighted Bergmanspaces on the open unit disc, an important progress for milestone 1 of task 1 in the network workplan.Task 2-Cauchy integrals, Capacities, Harmonic approximationIn this direction of research, following the complete characterization during the first year of network activityof planar Cantor sets of zero analytic capacity by J.Mateu, X.Tolsa and J.Verdera from team 3 in J. A.M.S.8

16 (2003), 19-28, and their partial results on semiadditivity in Contemp. Math. 320 (2003), 19-28 , X.Tolsasolved in the summer 2001 two outstanding problems by showing that the Ahlfors analytic capacity and thecontinuous analytic capacity are semiadditive. The first result, published in Acta Math. 190 (2003), 105-149shows in particular that the union of two sets which are removable for bounded analytic functions is alsoremovable for bounded analytic functions. The second result, published in Amer. J. Math. 126 (2004), 523-567 also solves the famous ”inner boundary conjecture” in rational approximation. Tolsa’s work also gives ageometric characterization of compact sets in the plane of zero analytic capacity (i.e. compact sets which areremovable for bounded analytic function), which answers a famous problem of Painlevé. After these crowningachievements, quickly expoited in the network training program, X.Tolsa obtained the prestigious Salem prizein 2003 and was one of the ten young recipients of the prize of the European Mathematical Society at 4ECMin june 2004 at Stockholm.These achievements obtained by the Barcelona team, notably by X.Tolsa, during the last year in the areaof analytic capacity and the Cauchy integral have continued during the last two years of network activity withother important results, which include the proof of the invariance of sets of zero analytic capacity by bilipschitztransformations in the preprint Bilipschitz mappings, analytic capacity, and the Cauchy integral (in the case ofplanar Cantor sets this bilipschitz invariance had been proved by J.Verdera (team 3) and J. Garnett in MathResearch Notes, 10(2003), 515-522). The central role of the Cauchy kernel within Calderon-Zygmund theory intwo dimensions is clarified in the preprints ”L 2 -boundedness of the Cauchy transform implies L 2 -boundedness ofall Calderon-Zygmund operators associated to odd kernels”, by X.Tolsa, and ”Estimates for the Cauchy integralover Ahlfors regular curves”, by X.Tolsa and M.Melnikov.Besides these decisive results concerning capacities and Cauchy kernel, which go far beyond the relatedresearch objectives of task 2 and milestone 1 of task 2, there were a lot of interesting results concerningapproximation and harmonic functions. S.Gardiner(team 4) and J.Pau (team 3) obtained in Illinois J. Math.47 (2003), 1115-1136 new results concerning the representation of functions on the boundary of a domain Ω bysums of Poisson, Green or Martin kernels associated to a subset E of Ω (this answers two questions of Hayman,and relates to milestones 1 and 2 of task 2). Also Gardiner and Hansen identified in Math. Ann. 323 (2002),41-54 the boundary sets where general harmonic functions may tend to infinity (milestone 2 of task 2), and theFarrel and Mergelyan sets of the Dirichlet space were classified by team 7.J. M. Anderson (team 4) and A. Hinkkanen used in Mathematika 48 (2001),301-304 Arakelyan’s theoremon holomorphic approximation to show that, if h and k are harmonic functions in the plane and there is apositive constant c such that |h + − k + | ≤ c in R 2 , then it need not follow that h − k is constant, which gives anegative answer to an open question related to quasi-regular maps in R n . Another important contribution toapproximation problems is given by F.Perez-Gonzalez (team 3) and A.Suarez in Int.Math.J.3 (2003), 795-810.Task 3-Function models and applications of operator theoryH.Langer(team 8), ACM Ran(team 2) and B. van de Rotten(team 2) obtained in Op. Th. : Adv. andAppl. 130 (2001), 235-254 new results about the existence of a pair of solutions of an algebraic Riccattiequation in the infinite dimensional case, using an invariant subspace approach. H.Langer(team 8) , A.Markus,V.Matsaev(team 9) and C.Tretter(team 8) obtained in Linear Algebra Appl. 330 (2001) 89 -112 seminal resultsabout the quadratic numerical range for block operator matrices, including numerical codes to compute it.Constrained von Neumann inequalities were obtained by C.Badea (team 1) and G.Cassier (team 6) in Adv.Math. 166 (2002), 260-297, with appealing applications to Fourier series.In the spectral theory of unbounded block operator matrices the investigations concentrated on delay equationsand the weakly coupled beam equations. The essential spectrum and semi-group generator could becharacterized and a complete semigroup treatment of delay equations with unbounded operators in the delayterm was given (A.Batkai, former postdoc at Vienna, and members of the Vienna and Amsterdam teams).9

Moreover, variational principles for operator functions and for block operator matrices could be proved, whichhave applications in various areas of mathematical physics. The study of selfadjoint and skewly selfadjoint blockoperator matrices led to interesting results about the angular operator representations of invariant subspaces,see the paper by H.Langer (team 8), A.Markus, V. Matsaev (team 9) and C.Tretter (team 8) in J.Func.An.199 (2003), 427-451.The study of the generalized Schur algorithm for Nevanlinna functions (the line case) was developed andimportant new results about the factorizations of J–unitary matrix polynomials, of operator representationsof generalized Nevanlinna functions and their applications in singular rank one perturbations of self-adjointoperators were obtained, see the paper by D. Alpay, A. Dijksma (team 2), and H. Langer (team 8) inLin. Alg.Appl. 387, 2004, 313-342. These results are also related via extension problems for symmetric operators toNevanlinna-Pick interpolation problems and as such to the results of Solution of a multiple Nevanlinna-Pickproblem by H.Langer (team 8) and the postdoc A.Lasarow (J. Math. An. and Appl. 293 (2004), 605-632),where the solutions of a Nevanlinna-Pick problem are described by means of orthogonal functions.Members of the Amsterdam and Tel-Aviv team also obtained a lot of new results on completeness andnon-completeness problems for non-selfadjoint operators, (noncompleteness of the eigenvectors and generalizedeigenvectors is closely related to the existence of the so-called small solutions) and on inversion of convolutionoperators. For the first topic a joint publication of I.Gohberg (team 9), M.A. Kaashoek and S.M. Verduyn Lunel(both team 2) is well underway. For the inversion see the paper by I. Gohberg (team 9), M.A. Kaashoek and F.van Schagen(both team 2) in The Erhard Meister Memorial Volume, OT 147 , Birkhauser Verlag, Basel, 2004,277–285.N. Nikolski proved with S.Treil in J.Anal. Math. 87 (2002), 415-421 that if U is a unitary operator whosespectral measure is not singular with respect to Lebesgue measure then there exists a rank one perturbation Tof U, whose spectrum does not contain the closed unit disc and whose resolvent has linear growth which is notsimilar to a normal operator, an important counterexample in the network program on similarity.Besides the joint contributions of contributions of A.Hartmann (team 1) and K.Seip (team 7) there was alot of progress concerning Toeplitz (or Hankel) operators. Sandra Pott (team 5) and Elizabeth Strouse (team1) found very recently a necessary and a sufficient condition for the product of two Toeplitz operators ona weighted Bergman space to be bounded involving the Berezin transform of the symbols of these Toeplitzoperators, and N.Nikolski (team 1) studied in a paper to appear in St Petersburg Math. J. the link betweencondition numbers (a notion useful in numerical analysis) and spectral condition numbers for Toeplitz operatorson l p (Z + ), 1 ≤ p ≤ +∞ (it is shown that theses phenomena involve a critical number k(p), equal to 2 for p = 1and p = +∞, equal to one for p = 2, whose value remains unknown for 1 < p < +∞, p ≠ 2). Hadamard-Schurmultipliers of Toeplitz type were characterized in St Petersburg Math. J. 15 (2003), 1-14 by Yu. Farforovskayaand L.N Nikolskaya, two senior female members from teams 10 and 1.Task 4-The invariant subspace problemLet ω be a weight on Z, i.e. a positive map for which the shift operator (u n ) n∈Z ↦−→ (u n−1 ) n∈Z is abounded invertible operator on the Hilbert space l 2 ω(Z) = {(u n ) n∈Z | ∑ n∈Z |u n| 2 ω(n) 2 < +∞}. The existenceand classification of nontrivial translation invariant (i.e. invariant for S and S −1 ) subspace of l 2 ω(Z) is an openproblem related to milestones 3 and 4 of task 4. The network coordinator (team 1) and A.Volberg (partly in team6), using the theory of asymptotically holomorphic functions, gave in Ann. Scient. Ec. Norm. Sup. 35 (2002),185-230 a complete classification of the translation invariant subspaces for a large class of dissymetric weightedHilbert spaces of sequences on Z (milestone 3 of task 4) and gave conditions on a weight ω which ensure thatevery nontrivial translation invariant subspace of the corresponding weighted Hilbert space contains a nonzerosequence (u n ) n∈Z such that u n = 0 for n > 0. The existence of nontrivial translation invariant subspaces in thecase where the spectrum of the shift is thick (milestone 4 of task 2) is still open (see St Petersburg Math. J. 1410

(2003), 251-271 for partial results).The translation invariant subspace for even weights had been solved positively by A. Atzmon (team 9), usinga ”moment principle”, before the beginning of network activity (see Ann. Inst. Fourier 51 (2001), 1407-1418for an extension of this result to a large class of Banach spaces of functions on l.c.a. groups), and he obtainedwith G.Godefroy (team 6), using a variational principle, a very general version of the moment principle: if T isa bounded operator on a real Banach space, and if there exists a bounded measure µ on R, x ∈ E \ {0} andx ∗ ∈ E ∗ \ {0} such that < T n x, y >= ∫ R tn dµ(t) for n ≥ 0, then T has a nontrivial invariant subspace.The so-called Brown approximation scheme plays a central role to construct nontrivial invariant subspaces.For example if T is an absolutely continuous contraction on the Hilbert space H for which the functionalcalculus h ↦−→ h(T ) is an isometry, this scheme shows that for every f ∈ L 1 (T) and every ɛ > 0 there existsx, y ∈ H such that < T n x, y >= ˆf(−n) for n ≥ 0, which implies that T has a very rich lattice of invariantsubspaces. After developing the Ansari-Enflo technique of minimal vectors in a paper to appear in Proc. AMSand giving a constructive proof of the fact that the classes A m,n of the theory of dual algebras are distinct,I.Chalendar (team 6), J.R.Partington (team 5) showed with R.Smith (team 5) that the existence of pairs (x, y)of elements of H such that < T n x, y >= ˆf(−n) ( n ≥ 0) for some specific f ∈ L 1 (T) does imply the existence ofnontrivial invariant subspaces for T (for example this is true for functions f ∈ L 1 (T) which agree a.e with thenontangential limit on the circle of a quotient of bounded analytic functions on the open unit disc) . In the samepaper they also establish for the first time a link between the Brown approximation scheme and the Hilbertspace version of the Atzmon-Godefroy moment theorem mentioned above. Other important contributions tothe Brown approximation scheme include a joint paper by I. Chalendar, B. Chevreau (team 1) and G. Cassier(team 6) in J. Op. Th. 50 (2003), 331-343 and a preprint by B. Chevreau related to milestone 1 of task 4.F. Jaeck (team 1) and S.C. Power (team 3) proved in a paper to appear in Proc.A.M.S. that free semigroupoidalgebras with finite graphs are hyper-reflexive, a result related to milestone 2 of task 4, and E.Abakoumov(team 6) and A.Borichev (team 1) constructed in J. Func. An. 188 (2002), 1-26 shift invariant subspaces witharbitrary indices in l p spaces. A.Atzmon (team 9) and B.Brive (team 1, postdoc at Tel-Aviv) characterizedsurjectivity of linear differential operators of infinite order with constant coefficients on weighted Hilbert spacesof entire functions with radial log-convex weight, and determined for some of these spaces the structure ofdifferentiation invariant subspaces. In another direction there were major advances concerning hypercyclicity(recall that T ∈ B(E) is said to be hypercyclic if the set {T n x} n≥0 is dense in E). Sophie Grivaux, a veryyoung Mathematician from Paris, proved that each operator on the separable Hilbert space can be written asthe sum of two hypercyclic operators. With F. Bayart (team 1), she points out in C. R. Acad. Sci. Paris, 338(2004), 703 - 708 the role of the unimodular point spectrum in hypercyclicity theory.Task 5-Geometry of Banach Spaces and applicationsE.Ricard, a young mathematician from team 6 showed in CRAS 331 (2000), 625-628 that the classicalHardy space H 1 , which has as well-known an unconditional basis, fails to have a completely unconditional one.He also constructed with T. Oikhberg in Math. Annalen 328 (2004), 229-259 an operator space on which everyendomorphism is the sum of a scalar multiple of the identity and a nuclear endomorphism. After obtaining anoncommutative version of Grothendieck’s theorem, G.Pisier (team 6) proved that the only quotients of thedirect sum of the row and column operator spaces that embed in a semifinite non-commutative L 1 space are R,C, R + C (the classical Gaussian case) and 4 other spaces built out of these first 3, and Q.Xu (Besançon, team6) developed complex interpolation of certain row and column operator spaces.Y. Raynaud (Paris 6) and Q.Xu obtained in J. Func. Anal. 203 (2003) important structural resultsconcerning subspaces of noncommutative L p -spaces over general Von Neumann algebras. Q. Xu obtainedwith C.Le Merdy (Besançon) and M. Junge several analogues of Littlewood-Paley and Stein inequalities forsectorial operators on noncommutative L p -spaces, after introducing a suitable notion of square functions for11

such operators,see CRAS Paris 337 (2003), 93-98. Le Merdy also showed in J. Austr. Math. Soc. 74 (2003)that under usual spectral conditions the sum of two operators having a bounded Mc Intosh H ∞ -calculus on aAUMD space also has a bounded H ∞ -calculus (the AUMD class contains the L 1 -spaces and the UMD-spaces).Concerning nonlinear theory of Banach spaces, G.Godefroy (team 6) proved with N. Kalton in Studia Math.159 (2003), 121-141 that when a linear quotient map to a separable Banach space has a Lipschitz right inverse,then it has a bounded linear right inverse, and R. Deville (team 1) gave with N. Ghossoub in Handbook ofthe geometry of Banach spaces, Vol. I, 393–435, North-Holland, Amsterdam, 2001 an up-to date overview ofminimization principle(this is related to milestone 3 of task 5).J.R. Partington (team 5) and I.Chalendar (team 6) applied to approximation in Archiv. Math. 78 (2002),223-232 results on real interpolation for Hardy spaces on circular domains. C.Dyakonov (teams 3 and 10)characterized in Math.Res. Letters 10 ((2003), 717-728 the extreme points of the unit ball of the space ofpolynomials of given degree living on the circle or on a real segment and endowed with the sup-norm. S.Kislyakov(team 10) introduced in Studia Math. 159 (2003), 277-290 a new ”weak” condition of BMO-regularity forcouples (X, Y) of lattices of measurable functions on the circle and showed that this condition still ensuresnice interpolation properties for the couple of the corresponding Hardy-type spaces. Also S.Kysliakov and hisgraduate student D.Anisimov established in Double singular integrals, interpolation and correction, to appear inAlgebra i Analiz interpolation and correction results for some function spaces defined in terms of certain ”doublesingular integral operators”, which apply to many classical operators of harmonic analysis (for instance, theHardy-Littlewood square function).Task 6-Convex geometry,concentration of measuresG. Aubrun and M. Fradelizi, from team 6, proved in Arch. Math. 82 (2004), 282-288 that the sphericalcaps are the only spherical convex bodies which remain spherically convex under the action of all 2-pointssymmetrizations (Schneider’s conjecture). Also M.Fradelizi, with D. Cordero and B. Maurey (also from team6) recently verified an inequality of multiplicative concavity conjectured by Banaszczyk relative to the gaussianmeasures of dilations of any given centrally symmetric convex set (the so-called “(B)-conjecture”).O. Guédon (team 6) and G. Paouris (a young researcher appointed by the Paris-6 node) studied in a jointpaper the concentration of mass of the unit ball of the finite dimensional Schatten class Sp n (with respect to theHilbert-Schmidt norm), in relation to a general conjecture on the comparison of q-th moments associated withcentrally symmetric convex bodies.A. Pajor (team 6) and M. Milman (team 9) proved in Studia Math. 159 (2003), 247-261 new resultsconcerning the regularization procedure of arbitrary star body obtained by cutting by random half-spaces,showing that the resulting convex body has (with large probability) better regularity properties. For examplecutting with suitable n/2 half spaces a l n 1 ball of diameter of order √ n containing the standard euclidean ballone obtains a body with (absolutely) bounded diameter and still containing the unit ball. This method of globaltype permits to recover most of the classical results in local theory, and relates to milestone 1 of task 6.After solving the duality problem for metric entropy in the central euclidean case V. Milman (Team 9),A. Artstein (former predoc at team 6 and graduate student of team 9) and S.Szarek (team 6) introduced withN.Tomczak-Jaegermann the notion of ”convexified separation”: given a set K and a symmetric convex body B afinite sequence (x 1 , ..., x N ) of elements of K is said to be B-convexly separated if (x j + int(B) ∩ conv{x i } i

FINAL REPORT -Part A- Research resultsA2 . Joint Publications and PatentsWe present here an provide a copy of five most significant joint publications which are considered to havehad a high impact . Young researchers are indicated in bold, and female authors are indicate with a * sign.List of five selected joint papers[1] S.Gardiner (team 4) and J.Pau (team 3), Approximation on the boundary and sets of determination forharmonic functions, Illinois Journal of Mathematics 47 (2003), 1115-1136.[2] H.Langer (team 8), A. Markus, V.Matsaev (team 9) and C.Tretter ∗ (team 8), Self-adjoint Block operatormatrices with non-separated diagonal entries and their Schur complement, Journal of Functional Analysis 199(2003), 427-451.[3] A.Hartmann (team 1) and K.Seip (team 7), Extremal functions as divisors for kernels of Toeplitzoperators, Journal of Functional Analysis 202 (2003), 342-362.[4] L.Nikolskaia ∗ (team 1) and Yu. Farforovskaia ∗ (team 10), Toeplitz and Hankel matrices as Hadamard-Schur multipliers St Petersburg Math. Journal 15 (2004) 1-14.[5] I.Chalendar ∗ (team 6), J.R. Partington(team 5), and R. Smith ∗ (team 5), L 1 -factorizations, momentproblems and invariant subspaces, preprint 2004.Comments on the five selected papersPaper [1]: Let E be a subset of a domain Ω in Euclidean space. In this paper J.Pau (postdoc at Dublin, fromBarcelona) and S. J. Gardiner (coordinator of the Dublin node), dealt with the representation, or approximation,of functions on the boundary of Ω by sums of Poisson, Green or Martin kernels associated with the set E, andwith the related issue of whether E can be used to determine the suprema of certain harmonic functions on Ω.The results address several questions raised by W. K. Hayman, and relate to Milestones 1 and 2 of Task 2 ofthe Network Workplan [(the reference ] to the network is to be found at the end of the introduction).A BPaper [2]: Let A =B ∗ be a block operator matrix in a Hilbert space H = HD1 ⊕ H 2 , with boundedoperators A, B and D, where A and D are self-adjoint. It is well-known that if the spectra of A and D areseparated, e.g. d = max[σ(D)] < min[σ(A)] = a, then the interval (d, a) belongs to the resolvent set of A andmin[σA] ≤ d < a ≤ max[σA]. Moreover the spectral subspaces of A associated to [a, +∞) is angular : thissubspace is the graph of a contraction K : H 1 → H 2 (a similar property holds of course for the spectral subspaceassociated to (−∞, d]). The purpose of the paper [2] is to investigate the situation where the spectra of A and Dare not separated. For example if the operator A has spectrum on a closed interval ∆ ⊂ ρ(D) then the spectralsubspace associated to ∆ has an angular representation associated to an operator K which is in general definedonly on a subspace of H 1 and is no longer a contraction. If the interval ∆ ⊂ ρ(D) is half-open or open then theoperator K may be unbounded. The first Schur complement S 1 (λ) = A − λ − B(D − λ) −1 B ∗ corresponding to∆ plays an important role in this investigation. This paper should become a reference for further investigationsbecause the methods used can be extended to some situations where the operator A is not self-adjoint andhas unbounded coefficients. The notion of angular subspace plays a important role in many situations, see forexample the paper ”Invariant Subspaces of Infinite Hamiltonians and Solutions of the Corresponding RicattiEquations”, by H. Langer (team 8), A.C.M. Ran (team 2) and B. Van de Rotten (team 2) devoted to algebraicRicatti equations arising from Control Theory. This work is related to milestone 2 of task 3 of the NetworkWorkplan.Paper [3]: Let E be a Banach space of analytic functions on the unit disc, and assume that the evaluationmaps f ↦−→ f(z) are continuous with respect to the norm of E for |z| < 1. If F is a closed subspace of E13

Last periodic report- Part B - Comparison with the joint programme of work(AnnexI of the contractLast periodic report -B1 Research objectivesA detailed discussion of the progress during the four years of network activity towards the research objectivesset down in annex 1 of the contract will be given in section B.1 of the final report. The fourth year of networkactivity confirmed the continued relevance of most of these project objectives. The papers [2] , [3], [4], [5] ofthe list provided in section A.2 for task 1 are important contributions to interpolation in spaces of holomorphicfunctions, and the works of Dussau, Perez-Gonzalez and Rattya mentioned in section A.1 of the present reportare contributions to sampling. No progress concerning the characterization of ”inner-outer” functions in termsof boundary behavior was reported, but a joint work by Borichev and Lyubarskii deals with the maximal decayalong appropriate sequences for various spaces of holomorphic functions. The central role of the Cauchy kernelwithin Calderon-Zygmund theory is clarified in two papers by Melnikov-Tolsa and Tolsa, and Gardiner solvedwith Pau and with Gustafsson, both postdocs in team 4, several problems concerning approximation of functionson the boundary of a domain by sums of classical kernels and existence of a smooth potential on a C 1 domainwith prescribed normal derivatives at the boundary. Papers [10], [12], [13], [14] for task 3 in section A.2 representan array of various significant contributions to the theory of Toeplitz and Hankel operators, and an importantjoint publication on completeness and non-completeness problems for non-selfadjoint operators by I.Gohberg(team 9), M.A. Kaashoek and S.M. Verduyn Lunel (both team 2) is well underway. New applications of theBrown approximation scheme and a link between this scheme and the Atzmon-Godefroy moment theorem aregiven in paper [3] for task 4 in section A.2. G.Pisier (team 6) characterized quotients of the direct sum of rowand columns of operator spaces which embed into a semifinite noncommutative L 1 -space, and Q.Xu developedcomplex interpolation of certain row and column operator spaces. In convex geometry the notion of ”convexseparation” and the related ”convex packing number” introduced in paper [2] for task 6 in section A.2 shouldplay in the future an important role in complexity and optimization theory.Last periodic report- B.2 Research methodThe interplay between Function Theory, Operator Theory, Geometry of Banach spaces and Convex Geometry,and the research of application of these theories to other branches of Mathematics was the basis of themethodology described in Annex 1 of the Contract. This methodology has been working well during the fouryears of network activity: tools as different as almost holomorphic functions, the Brown approximation scheme,variational principles and positive definite functions were used in teams 1, 2 and 9 to study translation invariantsubspaces of weighted l p spaces of sequences on Z. The interplay between complex function theory and operatortheory plays a central role in all the applications of operator theory developped by teams 2, 8 and 9, and theinterplay between Operator Theory and Geometry of Banach spaces was the basis of the progress reportedby team 6 concerning Operator Spaces and noncommutative Functional Analysis. Also the interplay betweenconvex geometry and complex analysis played a key role in the proof of the local dimension-free estimates forsublevel sets of analytic functions mentioned in B1. The Geometric notion of curvature of a measure introducedsome years ago by Melnikov and Verdera played a central role in the breakthroughs concerning analytic capacitiesperformed in team 3, and probabilistic methods also played an important role in several results obtainedin the network in function theory (team 3), harmonic analysis (team 1), problems of concentration of measure(teams 5 , 6 and 9) and noncommutative Functional Analysis (team 6). After four years of network activity15

it becomes possible to say more precisely that the interplay between Function Theory, Geometry, OperatorTheory, Geometry of Banach spaces, Convex Geometry and Probabilistic methods was the basis of the networkmethodology.Last periodic report- B.3 Work PlanBreakdown of tasksIn the following table, we compare for each task the teams expected to be involved in Annex I of the contractand the teams who reported progress made during the four years of network activity.task n o task title coordinator teams expected contributing teamsto contribute before 31/ 05 / 20041 Hardy and Bergman spaces, team 7 1, 3, 4, 7, 9, 10 1, 3, 4, 7, 9, 10Interpolation2 Cauchy Integral, Capacities, team 4 1, 3, 4, 7, 10 3, 4, 5, 6, 7, 10Harmonic approximation3 Function models and applications team 2 1, 2, 5, 7, 8, 9, 10 1, 2, 3, 5, 6, 7, 8, 9, 10of Operator Theory4 The invariant subspace team 9 1, 6, 7, 9 1, 2, 5, 6, 7, 8, 9problem5 Geometry of Banach spaces team 10 1, 5, 6, 7, 9, 10 1, 5, 6, 7, 9, 10and applications6 Convex Geometry, team 6 6, 9 5, 6, 9Concentration of measureWe thus see that the contributions of the team went along as expected. Teams 5 and 6 made unexpectedcontributions to task 2, with papers on approximation on Hardy spaces. Notice also that G.Blower from team5 contributed to the theory of concentration of measure. As could be observed at the network workshop oninvariant subspaces held at Leeds in July 2003, task 4 should be renamed ”Invariant subspaces”: the invariantsubspace problem is still open for all reflexive separable Banach spaces, but a lot of progress related to invariantsubspaces has been made in various directions. The so-called angular invariant subspaces for block operatormatrices play an important role in the solution of some algebraic Riccatti equations arising from operator theory,the Brown approximation scheme has been applied to noncontractive bilateral shifts, and decisive progresson hypercyclicity has been made by S. Grivaux, a (very) young Mathematician from the Paris team, whoshowed in particular that every bounded operator on the Hilbert space can be written as the sum of twohypercylic operators, a result which seemed out of reach at the beginning of network activity. I. Chalendarand J.R. Partington also investigated during the last year of network activity new applications of the Brownapproximation scheme, see paper [5] in part A.2 of the Final Report.Schedule and milestonesAs indicated above, the expected schedule was globally respected, with outstanding breakthroughs goingfar beyond expectations concerning analytic capacities in team 3 (most of them were published in the last yearof network activity). Some milestones were pointed out in Annex 1 of the contract to serve as a sampling ofthe progress of the network workplan. We reproduce in the following table the progress reported about themilestones for which the expected duration of research was 48 months (the milestones have a 2-digit numeration,16

where the left number gives the task number: for example milestone 2.3 is the third milestone for task 2).These milestones still give a reasonable sampling set to evaluate the progress on the network workplan withtwo exceptions: milestone 4.2 has been reformulated as: ” Hyper-reflexivity and description of the lattice ofinvariant subspaces for noncommutative algebras more general than the Fourier binest algebra”, and exampleswhere Pisier’s similarity numbers satisfy 3 < d < +∞ are still to be found.Progress concerning the Network Milestones (case of expected duration of 48 months)Milestone Milestone Duration of research Progress reported beforenumber title mentioned in 31/05/2004Annex 1 of contract1.1 Existence of zero-free noncyclic vectors 48 months Important progress reported infor all weighted Hardy spacesthe log-convex case in joint paperby teams 1, 6, 7. General case still resists2.1 Applications of capacities to approximation 48 months Publication of the solutions of Painlevé’ sproblems in spaces of analyticand harmonic functionsproblem and of the inner boundary conjectureby Tolsa; bilipschitz invariance of analyticcapacity proved in team 3; important progresson harmonic approximation reportedfrom team 43.2 Use of the band method to 48 months Progress reported from teams 2 and 9describe the set of all solutionsfor some Takagi typeinterpolation problems4.2 Hyperreflexivity and description 48 monthsof the lattice of invariant subspacesJoint paper by Jaeck (team 1) and Powerof noncommutative algebras more general(team 5) gives a significant contribution tothan the Fourier binest algebra this revised version of milestone 4.2.4.4 Reflexivity properties of the shift 48 months Partial results obtained in team 1;operator on lω 2 (Z)in the case of thick spectrumthe general case still resists.The Apostol paradox provides functionsanalytic inside and outside the discfor which the product of boundary valuesvanishes.5.1 Examples where Pisier’s similarity 48 months No progress reported so farsatisfies 3 < d < +∞17

Last periodic report- B.3 Research effort of the participantsIt is difficult to quantify precisely in person-months the research effort of the scientific staff from the differentteams, and so the numbers given in column 2 of the table below are estimates. We give in bold the numberscorresponding to the four years of network activity (for the third column it is the number of individuals forwhom a concrete contribution to the research objectives of the network was reported), and we give betweenbrackets what was indicated in Annex I of the contract for the four years of network activity. The contributionof predoctoral students is not quantified (it was not quantified either in Annex I of the contract).Concerning young researchers appointed by the network we also give in bold the numbers corresponding tothe four years of network activity, and we give between brackets what was deliverable according to Annex Iof the contract. The appointment of young researchers will be discussed in more detail in section B.5.2 of thepresent report.Professional effort on the network projectParticipant Young Researchersresearchers Researchers to be contributingfinanced by the financed from other to the projectcontract sources (number of(person month) (person- month) individuals)1. UB1 41 (36) 290 (294) 22 (24)2. VUA 28 (24) 185 (180) 10 (12)3. UAB 39 (36) 180 (176) 17 (16)4. UCD 28 (24) 170 (160) 12 (11)5. ULeeds 24 (24) 196 (192) 12 (14)6. UPMC 37 (36) 300 (312) 24 (31)7. NTNU 24 (24) 160 (160) 12 (12)8. TU Vienna 26 (24) 120 (120) 8 (8)9. TAU 24 (24) 110 (112) 10 (10)10. POMI 0 (0) 120 (112) 8 (8)TOTAL 271 (252) 1831 (1838) 135 (134)18

Final Report- B.1 Research achievementsThe interplay between Function Theory, Operator Theory, Geometry of Banach spaces and Convex Geometry,and the research of application of these theories to other branches of Mathematics was the basis of themethodology described in Annex 1 of the Contract. The four periodic and the midterm review reports showthat this methodology has been working well during the four years of network activity. The Geometric notionof curvature of a measure introduced some years ago by Melnikov and Verdera played a central role in thebreakthroughs concerning analytic capacities performed in team 3, and probabilistic methods also played animportant role in several results obtained in the network in function theory (team 3), harmonic analysis (team1), problems of concentration of measure (teams 5 , 6 and 9) and noncommutative Functional Analysis (team6). After four years of network activity we can say more precisely that the interplay between Function Theory,Geometry, Operator Theory, Geometry of Banach spaces, Convex Geometry and Probabilistic methods was thebasis of the network methodology.Concerning the breakdown of tasks a detailed table is given in section B.3 of the last periodic report. Teamscontributions went along as expected concerning tasks 1 and 5, team 5 made unexpected conditions to task2 to which team 1 did not contribute directly as planned (but contributions on Toeplitz operators on Hardyor Bergman spaces have a link to the Cauchy integral), teams 3 and 6 made unexpected contributions totask 3 and G. Blower, from team 5, also contributed to concentration of measure (task 6). Task 4 should berenamed Invariant subspaces instead of ”The invariant subspace problem,” because the description of some orall invariant subspaces of concrete classes of operators played a role in several directions of research in thenetwork, see for example paper [2] in section A.2 of the present report and the programme of the Workshopon invariant subspaces organized at Leeds in July 2003, which is available on the network homepage. Teams 2,5, 8 joined the expected teams 1, 6, 7, 9 to contribute to this task, which benefited from the emergence of anew generation of brillant young mathematicians (F. Bayart, team 1, Sophie Grivaux, team 6) who solved longstanding questions concerning hypercyclicity.An important project objective in function theory consisted in the characterization of interpolating andsampling sequences in one and multidimensional situations, which was done for the Paley-Wiener space byOrega-Cerda and Seip, for the Bloch space by Boe and Nicolau, and for functions of restricted growth and forthe Nevanlinna and Smirnov classes by Hartmann-Massaneda and Hartmann- Massaneda- Nicolau, see papers[1], [5], [6], [7] of section A.2 of the midterm review report. Also A.Nicolau and K.Dyakonov obtained resultson interpolation by nonvanishing functions, a direction of research which might play an important role in thefuture. There was no direct progress on inner-outer type factorization in Bergman and related spaces, but theconstruction of non z-cyclic functions without zeroes for a large class of weighted Bergman spaces paves the roadto find in this very general context a suitable analog for singular inner functions. There were no new invertibilitycriteria for Toeplitz operators reported so far, but Hartmann-Seip found with Sarason a nice characterizationof surjective Toeplitz operators with nontrivial kernel.Understanding of capacities in metric and geometric terms went far beyond what was expected, with thesolution by Tolsa of Painleve’s problem, of the inner boundary conjecture, and with his proof of the semiadditivityof the analytic and continuous analytic capacities. Some other important issues in harmonic approximationand behavior of harmonic functions at the boundary were adressed by teams 3 and 4.Concerning research objectives in Operator theory, little progress was reprted about function models butthere was a lot of progress concerning Hankel and Toeplitz operators on Hardy and Weighted Bergman spaces,a topic at the crossroads between operator theory and function theory. Several papers by teams 2, 8 and 9developed operator theoretical methods to analyze problems arising from concrete classes of integral differentialand delay equations and descibed the spaces spanned by generalized eigenvectors for nonselfadjoint operatorslinked to delay equations. The Brown approximation scheme was applied to single contractions by teams 1and 6, to n-uples of contractions in team 1 and to translation invariant subspaces of l 2 ω(Z) in team 1 (the19

latter situation does not involve contractive operators). Function theoretical tools as the minimum principlefor asymptotically analytic functions were applied as planned by teams 1 and 6 in a joint paper mentioned insection A.1 of the present final report.Concerning Geometry of Banach spaces and Convex Geometry important new results were obtained in team6 on operator spaces (for example the noncommutative version of Grothendieck’s theorem due to Pisier) andthe theory of noncommutative L p -spaces was extended to more general underlying von Neumann algebras bymathematicians from Besançon and Paris (both in team 6). Important new applications of the principle ofconcentration of measure were developed in team 6 by Talagrand, and new approximation algorithms of convexbodies by polytopes were obtained in joint works by team 6 and 9. Rather surprisingly the main application ofvariational principles happens the be the spectacular joint result on invariant subspaces by A.Atzmon (team 9)and G.Godefroy (team 6) mentioned in section A.1.We summarize in the following table the progress reported about the various milestones indicated in Annex1 of the contract (the milestones have a 2-digit numeration, where the left number gives the task number:for example milestone 2.3 is the third milestone for task 2). The global situation observed from this point ofview is also satisfactory: milestone 2.2 has been completely clarified, and Tolsa’s results go much beyond whatwas suggested in milestone 2.1. Important partial progress is reported concerning milestones 1.2, 2.3, 3.1, 3.2,4.1, 4.2, 4.3, 4.4 and 6.2 (milestone 4.2 was reformulated in the third periodic report, and progress concerningmilestone 6.2 is due to a work of M. Talagrand which was unfortunately not exploited in the training program).Milestone 5.3 was completely clarified from outside the network, and even if no direct progress was reportedfor milestones 1.3, 5.2 and 6.1 some analogous problems were solved during the network activity. Milestone 6.1still seems out of reach. For milestone 3.3 the story is different: the µ-synthesis problem still resists, but theresearch on this problem by J. Agler and N.J.Young (team 6) led to the paper The hyperbolic geometry of thesymmetrised bidisc, to appear in J. Geometric Analysis, which attracted a lot of attention among experts incomplex analysis in several variables.Progress concerning the Network MilestonesMilestone Milestone Duration of research Progress reported beforenumber title mentioned in 31/05/2004Annex 1 of contract1.1 Existence of zero-free noncyclic vectors 48 months Important progress reported infor all weighted Hardy spacesthe log-convex case in joint paperby teams 1, 6, 7. General case still resists1.2 Characterization of complete interpolating sequences 24 months Done by team 7 in important casesin weighted Hilbert spacesof entire functions1.3 Characterization of discrete random fields 24 months No progress reported so far ;for which entire functions with zeroesresults on the distribution of zeroes forin the field belong almostsome other classes of analyticsurely to the Fock space functions reported from team 92.1 Applications of capacities to approximation 48 months Painlevé’ s problem andproblems in spaces of analyticand harmonic functionsand the inner boundary conjecturewere solved by Tolsa;bilipschitz invariance of analyticcapacity also proved in team 3;important progresson harmonic approximation reportedfrom team 420

Progress concerning the Network Milestones (continued)Milestone Milestone Duration of research Progress reported beforenumber title mentioned in 31/05/2004Annex 1 of contract2.2 Characterization of exceptional sets 24 months Done by team 4at the boundaryfor harmonic functions2.3 Computation of best constant inequalities 24 months Progress reported from team 7 and 10for conjugate harmonic functions3.1 Explicit computation of 24 months Tests for similarity to a normalthe characteristic function operator obtained in team 1;for new classes of contractions,counterexample involving a rank oneand obtain resolvent tests forperturbation of a unitary operatorsimilarity to a normal operator also constructed in team 13.2 Use of the band method to 48 months Progress reported from teams 2 and 9describe the set of all solutionsfor some Takagi typeinterpolation problems3.3 Use of the Agler-Young operator- 24 months the µ-synthesis problemtheoretical method to make progress onThe µ-synthesis problem in engineeringstill resists but this researchled to unexpected applications to complexanalysis in several variables4.1 Reflexivity for pairs of commuting 24 months Progress reported from team 1contractions with dominantHarte spectrum4.2 Hyperreflexivity and description 48 monthsof the lattice of invariant subspacesJoint paper by Jaeck (team 1) and Powerof noncommutative algebras more general(team 5) gives a significant contribution tothan the Fourier binest algebra this revised version of milestone 4.2.4.3 Complete description of translation biinvariant 24 months Decisive existence results of translationsubspaces of lω(Z) 2 for a large class ofbiinvariant subspaces have beennonincreasing weights not bounded awayobtained by team 9 for even weights;from 0a joint paper by teams 1 and 6 gives results inthis direction for weights equal to 1 on Z +4.4 Reflexivity properties of the shift 48 months Partial results obtained in team 1;operator on lω 2 (Z)in the case of thick spectrumthe general case still resists.The Apostol paradox provides functionsanalytic inside and outside the discfor which the product of boundary valuesvanishes.5.1 Examples where Pisier’s similarity 48 months No progress reported so farnumber satisfies 3 < d < +∞5.2 Comparison between regularity and λ-regularity 24 months No direct progress so far. Relatedfor sums of operators in U.M.D. spaces results have been obtained in team 65.3 Formula for the Frechet subdifferential 24 months Partial results have been obtainedof the sum of two positive lowerin team 1; the problem was thensemicontinuous functions in smoothsolved from outside the networkBanach spaces6.1 Improvement of the Kannan-Lovacz 24 monthsestimate O(n 2 ) for the flatness ofteams 6 and 9 obtained jointlyn-dimensional bodiesimportant related results6.2 Explicitation of a direct link between 24 months Important results in this directionthe Sherrington-Kirkpatrick model were obtained in team 6and questions related to the groupof isometries of R n ,n very large21

A table given in section B.3 of the last periodic report shows the the number of researchers involved andthe person-months of research efforts developed in the ten teams correspond to what was expected. The totalnumber of person months of appointment is 271, 15 months more than was written in Annex 2 of the contract.This is due to the fact that some teams who were facing lower appointment costs provided more person-monthsof appointments.22

Last periodic report- B.4 Organisation and managementB.4.1After the initial Bordeaux meeting (June 12, 2000) at the beginning of network activity, the St Petersburgmeeting (May 12, 2001), the Biarritz meeting (May 5, 2002), the Paris meeting (November 20, 2002) and theTenerife meeting (May 24, 2003), which were organized during the first, second and third annual conferencesof the network and just after the midterm review meeting, there was a meeting of network coordinators in thefinal year of network activity, organized on May 2, 2004 during the last annual conference of the network atDalfsen. These meetings played a central role in network management: decisions were made about appointmentsof young mathematicians, the preparation of annual reports was discussed, and the location and the topics ofthe morning lectures of the next network annual conference were choosen. Also these meetings were the placeswhere decisions are made about the Conferences to which the network would officially participate and wherethe network workshops on specific subjects were planned.The network strategy for dissemination of results and communication, besides visits, conferences and workshopsconsisted in using the network homepage. The four annual reports and the midterm review report areavailable there (dvi format), there is a link with the homepages of the four annual meetings, the three postdocworkshops and four of the specialized workshops which occured during the final year of network activity. Thismakes for example the program and the abstracts of the main lectures and the short talks of the third andfourth annual conference available on line (we put also on line a copy of the transparencies used by some ofthe main lecturers). A database of all the joint preprints and papers by members of different teams which wereproduced since the beginning of network activity has benn organized on the homepage: a list of these papers andpreprints is available in the annual reports, and there is now a link on the network homepage to the homepagesof network participants and to the homepages of the teams which have one.All network participants were regularly informed about network activities by messages from the principalcoordinator (using the alias of all network participants). The slides used by the network coordinator at hispresentation conference of the network at 4ecm (Stockholm, June 28-July 2, 2004, one month after the end ofnetwork activity) are also available on line.During the last work of network activity, the network funded the participation of some network members atseveral international conferences where results of the network research were presented.1. International Workshop on Operator Theory and Applications (IWOTA 2003), Cagliari, Italy, June 24–28,2003, team 1 (2 participants), team 2 (5 participants), team 6 (1 participant) and team 8 (5 participants).2. Nevanlinna Colloquium, Jyvaskila, June 2003, M.Melnikov (team 3).3. The Coifman-Meyer Conference, June 2003, Orsay, J.Verdera (team 3).4. 3d Internat. Workshop on Convex geometry - Analytical aspects, Cortona (Italy), June 8-14, 2003, A.Pajor, M.Fradelizi (team 6).23

5. Orlicz Centenary Conference on Function Spaces, Poznan (Pologne), July 21-25, 2003, Y. Raynaud, (team6).6. Conference on Mathematical Analysis in honor of V.P. Havin, St. Petersburg, August 15-20, 2003,A.Borichev, L. Nikolskaya, N. Nikolski (team 1) J. M. Anderson (team 4), E.Abakumov, Q. Xu (team 6)7. Journées Complexes du Sud, Carcassonne, November 17-19, 2003, J.Bruna (team 3).8. 4th Workshop Operator Theory in Krein Spaces and Applications, December 12-14, 2003, TU Berlin, A.Dijksma (team 2) H.Langer, M.Langer, A.Luger, C.Tretter (team 8)9. Colloquium on Operator Theory, Vienna, March 4 – 6, 2004. Participants: A. Dijksma, M.A. Kaashoek,A.C.M. Ran, H.V.S. de Snoo, (team 2), H.Langer, M.Langer, A.Luger, A.Wagenhofer, M.Winklmeier,H.Woracek, (team 8).The references of many homepages of network members can be found in the third periodic report, availableon the network’s homepage : http://maths. leeds.ac.uk/pure/analysis/rtn.html. Only one new homepage wasreported in the last year of network activity from team 5, the page of Z.A. Lykova (Newcastle, team 5):http://www.ncl.ac.uk/math/staff/profile/z.a.lykovaB.4.2.The third pre/postdoc workshop of the network took place at Paris on January 22-23, 2004. Ten youngmathematicians (Y. Ameur and B. Brive, postdocs in team 9, X.Dussau, team 1, former postdoc in team 3, S.Grivaux, team 6, M.Kopp, team 5, postdoc in team 1, K.Michels, former postdoc from team 5, A. Olofsson,team 7, former postdoc in team 2, D. Popovici, postdoc in team 8, former postdoc in team 1, J.Rattya, postdocin team 3 and M.Smith, postdoc in team 2 and future postdoc in team) gave a one hour talk.The fourth network annual conference was held at Dalfsen (The Netherlands) from May 21 to May 26, 2004,with 50 participants from the network, representing the 10 teams and including 22 young mathematicians, andtwo participants exterior to the network, including V. Vasilevskii, from Mexico, an international expert onToeplitz operators on the Bergman space. Three series of morning lectures were given1. Singular Integrals and Capacities, four 45 mn lectures by J. Verdera, team 3, and four 45 mn lecturesby G.David, team 6.2. Delay equations and infinite dimensional systems, four 45 mn lectures by J. Partington, team 5,and four 45 mn lectures by S. Verduyn Lunel, team 2.3. Toeplitz operators on Bergman spaces, by N.Vasilevskii (University of Mexico), six 45 mn lectures.There were also 27 short lectures (25 mn) on all directions of research of the network.The network also organized four more specialized workshops during the last year of network activity.1. Workshop on invariant subspaces, Leeds, July 3-5, 2003, organized by I.Chalendar (team 6), andJ.Partington (team 5), 14 participants from the network, including seven young mathematicians, oneparticipant exterior to the network . Teams 1, 5, 6 and 9 were represented.24

2. Workshop on spaces of analytic functions and applications, Trondheim, July 2-4, 2003, 22 participantsfrom the network, 7 participants exterior to the network. Teams 1, 3, 7, 9 and 10 were represented.3. Workshop on operator theory and applications, Amsterdam, August 20-22, 2003, attended bymembers of teams 2 and 8.4. Belfast Functional Analysis Day, a network activity organized by M.Mathieu (team 4), with theparticipation of members of teams 1, 4, 5, 6.5. With the department of Mathematics of Universitat de Barcelona and Universitat Autonoma de Barcelonaand the Gelbart Institute of Bar-Ilan University, the network organized the Conference on Bergmanspaces and related topics in Analysis in honor of B.Korenblum at Barcelona, November 20-23, 2003, consisting in 16 invited lectures by known specialists of the area, many of them attached tothe network (N.Nikolskii, team 1, A.Nicolau, team 3, H.Hedenmalm, team 7, A.Atzmon, team 9 andM.Sodin, team 9). The members of the scientific commitee were J.Bruna (team 3), H.Hedenmalm (team7), B.Pinchuk (Bar-Ilan University), K.Seip (team 7) and K. Zhu (SUNY at Albany). In occasion of themeeting a number of researchers from the other network nodes visted Barcelona, including A.Borichev(team 1), E. Strouse (team 1), M. Anderson (team 4), D. Armitage (team 4), S.Gardiner (team 4), A.O’Farrell (team 4), D. Walsh (team 4), S. Buckley (team 4), E.Abakumov (team 6), and A.Stray (team7).B.4.3During the last year of activity the network funded partially or completely several short (one week) ormiddle-size (from 2 to 6 weeks) visits from one team to another one.H. Langer (team 8) and A. Luger (team 8) visited A. Dijksma (team 2) and M.A. Kaashoek (team 2)in November 2003. Z.Lykova (Newcastle, team 5) and N.J.Young (Newcastle, team 5) visited C.Badea andF.Vasilescu (Lille, team 1) for one week in April 2004. C.Badea visited Newcastle for one week in May 2004,and had there fruitful discussions with N.J. Young and M. Dritschel. J. Bruna (team 3) visited K.Seip (team 7)from January 15 to February 15, 2004 to discuss linear independence of time frequency translates. The networkalso partially funded visits of I.Chalendar (Lyon, team 6) to Leeds (team 5). I.Gohberg (team 9) visited M.A.Kaashoek (team 2) and S.M. Verduyn Lunel (team 2) in August 2003 and from April 16 to May 31, 2004. M.Mathieu (team 4) made a research visit to the Technical University of Vienna (team 8), October 10-14, 2003and to Université Lyon 1 (team 6), March 8-14, 2004, and J. Pau (team 3) made a research visit to UniversityCollege Dublin, March 14-23, 2004. The St Petersburg branch of the Steklov Institute(team 10) was visited byH.Hedenmalm for 3 weeks in July 2003, by L.Nikolskaia (team 1) for 2 weeks in January 2004, and by A. Volberg(team 6) in April 2004. A.Atzmon (team 9) visited Barcelona during 10 days besides attending the BarcelonaConference on Bergman spaces. A.Aleksandrov (team 10) and S. Kapustin (team 10) visited Trondheim whileattending the network workshop on analytic spaces, and A.Baranov (team 10) visited Barcelona in May 2004.V.Vasyunin (team 10) visited Bordeaux in April 2004 and he visited Trondheim and KTH Stockholm in May2004. The two weeks visits of Kislyakov (team 10) to Trondheim (team 7) in January-February 2004 andof S.Kapustin to KTH Stockholm in November 2003 were paid from sources exterior to the network, and A.Baranov (team 10) benefited from specific french support for a six months research visit at Bordeaux (team 1).We summarize this information about research visits in tabular form: each research visit totally or partiallyfunded by the network is indicated by the sign X, and each visit funded from other sources is indicated by thesign (X). The sign x means that some members of one team attended a network event organised by anotherteam (network annual conference or network workshop).25

From / to Team 1 Team 2 Team 3 Team 4 Team 5 Team 6 Team 7 Team 8 Team 9 Team 10Team 1 X X X (X)Team 2 x x x x x x x X X x X X xTeam 3 x x x x XTeam 4 x X x xTeam 5 x X x X xTeam 6 x x x x X x X x xTeam 7 x x X x x X X (X) (X)Team 8XTeam 9Team 10 X X26

Final report B.2 Overall organization and managementThere were 6 meetings of network coordinators from June 1, 2000 to May 31, 2004. The first one was heldon June 12, 2000 at Bordeaux and the second one was held on May 12, 2001 at Saint Petersbourg, the daybefore the beginning of the first annual conference of the network. The third meeting occurred on May 5, 2002at Biarritz during the second network annual conference, the fourth one occured at Paris on November 20,2002 after the midterm review meeting, the fifth one occured at Tenerife on May 27, 2003 during the thirdnetwork annual conference and the last one occured on May 2, 2004 at Dalfsen, in the Netherlands, duringthe fourth network annual conference. These meetings played a central role in network management: decisionswere made about appointments of young mathematicians, the preparation of annual reports were discussed,and the location and the topics of the morning lectures of the next Annual Conference were chosen. Also thesemeetings were the place where decisions were made about the Conferences to which the network would officiallyparticipate and where the network workshops were planned.All network participants were informed about network activities by messages from the principal coordinator(using the alias of all network participants). A global view of network activity has been provided on the networkhomepage: the scientific part of the annual reports was available on line, as well as the schedule and abstractsof talks at the network annual conferences, at the network annual postdoc workshops and at the networkspecialized workshops. In the second half of network activity links were established to personal or researchgroup’s homepages within the network, thus providing links to the preprints mentioned in the periodic reports.The slides of several series of morning lectures at the third and fourth annual conferences are also available online on the network homepage, as well as the slides used by the network coordinator at Stockholm to presentthe network activities on July 2, 2004 during the morning session of 4ecm devoted to networks.The network organized four annual conferences, of a duration of 5 working days, with series of morninglectures on specific topics in the morning and short talks on all research directions of the network in theafternoons (the contents of these morning lectures will be discussed in section B3 of this final report). Also itappeared after one year of network activity that there was a need to organize each year a specific workshopfor the young mathematicians appointed by the network (or formerly appointed by the network), where eachof them would be given the opportunity to give a one hour talk. Three of these workshops were organized(the second one was organized just after the mid-term review meeting, allowing many young mathematiciansto attend the midterm review meeting and to discuss with the Brussels Officer).It was planned in the proposal to organize more specialized workshops on specific topics. This kind ofactivity developed on a large scale in the second half of network activity.We summarize in the following table these network activities. For the pre/postdoc workshop the quantityx + y (+z) gives the number x of speakers appointed by the network at the time of the meeting, the numbery of speakers who were future or former network appointees at the time of the meeting and the number z ofyoung speakers not appointed by the network. For the other activities when provided the quantity x (+y)indicates the number x of participants from the network and the number y of participants from outside thenetwork. Notice that the annual Belfast Functional Analysis day, which started in 1997, was integrated as aworkshop in the network activities in 2002 and 2003. The last of the events listed below was organized jointlyby the Department of Mathematics of Universitat de Barcelona and Universitat Autonoma de Barcelona, bythe Gelbart Institute at Bar-Ilan University and by the network, and it was dedicated to B. Korenblum. Manyof the 16 invited speakers were network participants, and 21 network participants came from outside Barcelona.The other workshops listed below were entirely organized by the network.27

Location Type of Organizer Task Participating Participantsand date activity teamsSt Petersburg annual conference team 10 all all teams 44 (+4)May 13-17 , 2001Biarritz annual conference team 1 all all teams 68 (+3)May 2-7, 2002Tenerife annual conference team 3 all all teams 67 (+14)May 21-26, 2003Dalfsen annual conference team 2 all all teams 50 (+ 2)May 1-7, 2004Bordeaux annual pre/ team 1 all all appointing 11 +1Jan. 18-19, 2002 postdoc workshopParis annual pre/ team 6 all all appointing 7+4Nov. 21-22, 2002 postdoc workshopParis annual pre/ team 6 all all appointing 7 + 2 (+1)Jan. 22-23, 2004 postdoc workshopVienna workshop team 8 3 team 2, 8, 9 14Jan. 18-19, 2001Trondheim workshop team 7 1 teams 1, 3, 7, 9, 10 22 (+7)July 2-4, 2003Leeds workshop team 5 4 teams 1, 3, 5, 7, 9, 10 13 (+1) speakersover 30 participantsJuly 3-5, 2003Amsterdam workshop team 2 3 teams 2, 5, 8, 9 21 (+9)Aug. 20-22, 2003Belfast workshop team 4 3 and 4 teams 4, 5, 6 ?Nov.16, 2002Belfast workshop team 4 3 teams 4, 5, 6Nov.15, 2003 21 (+5)Barcelona workshop team 3 1 and 2 teams 1, 2, 3, 4, 6, 7, 9Nov. 20-23, 2003 57 (+24)Besides the organization of its own conferences and workshops the network had a strategy of participation toconferences relevant to network research plan. The list of such Conferences was discussed at the meetings ofnetwork coordinators, with a few corrections between the meetings by email discussions. Priority was given toevents organised in network nodes. The following two tables give a list of these Conferences.1) Conferences not organized in network nodes 11 GDR CNRS 2101 was a French network, funded By the French National Research Center CNRS, to which most members ofteams 1 and 6 participated. The coordinator of this French network until December 31, 2003 was the network coordinator28

Conference Country Date of Conference Teams involved Teams involvedLocation Conference Title (with network funding) (with funding fromother sources)Orléans France 10/2000 Journées d’Analyse 3 and 5 1 and 6du GDR CNRS 2101 (funded by GDR 2101)Crete Greece 08/2001 Convex Geometry 6 and 9ConferenceOdense Denmark 08/2001 Balticon 1 and 6Umea Sweeden 06/2002 Analysis Conference 2 and 4Timisoara Romania 06/2002 Operator Theory 1, 5, 6 and 8ConferenceLuminy France 09/2002 Holomorphic Function 1, 6 (2 young) and 7 1 and 6Spaces and their (funded by GDR 2101)OperatorsCagliari Italy 06/2003 IWOTA 1, 2, 6, 8 9Jyvaskila Finland 06/2003 Nevalinna Colloquium 3(function theory)Orsay France 06/2003 Conference in honor 3of R.Coifman and Y.MeyerCortona Italy 06/2003 Workshop on 6Convex GeometryPoznan Poland 07/2003 Orlicz Centenary Conference 6on Function SpacesCarcassonne France 11/2003 Journées Complexes 3 1du SudBerlin Germany 12/2003 Operator Theory 2, 8in Krein Spaces2) Conferences located in network nodes, but not organized by the networkConference Country Date of Conference Teams involved Teams involvedLocation Conference Title ( network funding) ( funding fromother sources)Bordeaux, team 1 France 06/2000 IWOTA 2, 3, 5, 7, 8, 10 1 and 6(GDR 2101)Besancon, team 6 France 06/2000 Functional Analysis 5 1 and 6(GDR 2101)Summer SchoolSt Petersburg, team 10 Russia 08/2000 Analysis Conference 1Ambleside, team 5 England 09/2000 Operator Theory 1 and 6 5ConferenceAmsterdam, team 2 Netherlands 10/2000 Operator Theory 1 and 8ConferenceNewcastle, team 5 England 06/2001 Conference in honor 1 and 4of B.E.JohnsonSt Petersburg, team 10 Russia 08/2001 Analysis Conference 1Lyon, team 6 France 10/2001 Journées d’Analyse 1 and 6GDR 2101 (GDR 2101)Belfast, team 4 U.K. 11/2001 Belfast Functional 5 4Analysis DaySt Petersburg, team 10 Russia 08/2002 Analysis Conference 1Besançon, team 6 France 09/2002 Journées d’Analyse 3 1 and 6GDR 2101 (GDR 2101St Petersburg, team 10 Russia 08/2003 Analysis Conference 1, 4, 6Lens, team 6 France 09/2003 Journées d’Analyse 1 and 6GDR 2101 (GDR 2101)Vienna, team 8 Austria 03/2004 Colloquium on Operator Theory 2, 829

The following table indicates the number of visits from one team (horizontal entries) to another (verticalentries). Forty seven of such visits, funded totally or partially by the network, were reported between June 1,2000, and May 31, 2004. The starred numbers correspond to the sixteen other visits, directly related to networkactivities but funded from other sources, reported for the same period (this proportion of three visits fundedby the network compared to one visit funded from other sources is not unusual, since the network was intendedto stimulate already existing collaborations). The details about these individual visits can be found in the fourperidic reports (the first, second and third periodic reports are available on the network homepage).team N o 1 2 3 4 5 6 7 8 9 101 0 0 0 6 2 0 0 0 1 + 2 ∗2 0 0 0 0 0 2 5 + 1 ∗ 22 03 1 ∗ 0 0 0 0 0 0 1 04 0 0 2 0 0 0 0 0 05 1 ∗ 0 0 0 3 + 2 ∗ 0 0 0 06 1 +1 ∗ 0 0 1 7 + 1 ∗ 0 0 0 07 1 ∗ 0 3 0 0 0 0 0 2 + 2 ∗8 0 1 +2 ∗ 0 1 0 0 0 0 09 1 0 1 0 0 1 ∗ 0 0 010 5 + 1 ∗ 0 1 0 2 ∗ 1 ∗ 1 0 0The combination of these five activities (annual conferences, annual pre/postdoc workshops, specializedworkshops, participation to Conferences not organized by the network and mutual visits) and the fact thatnetworking activities increased in the second half of network duration (26 visits funded by the network afterthe mid-term review meeting compared to 21 before, 6 specialized workshop after the mid-term review meetingcompared to 2 before) led to a steady flow of joint papers by members from different teams produced duringthe four years of network activity. Transfer of knowledge and training greatly benefited from the infrastructureof the network: for example a first course on capacities was provided by team 4 at the first annual meeting,Tolsa detailed his breakthroughs in this area at the second annual meeting, and a new overview of the subjectwas presented by teams 3 and 6 at the last annual meeting. This networking activity, which took fully intoaccount the complementarity between the teams, indeed greatly stimulated progress in the three main directionsof research of the network at a Community level.30

Last periodic report- B.5 TrainingB.5.1Vacant positions have been publicised as usual through announcements put on the network homepage andsent to all network participants using the network alias. They were also sent to more than one hundred peoplein Banach algebras and Operator Theory through an alias run from California by M.Thomas and diffused inRomania through personal contacts of network members in several Romanian Universities.B.5.2 The following table indicates the recruitment of young researchers.The numbers on bold show in personmonthsthe number of person-months young researchers financed by the contract, and the numbers betweenbrackets indicate what was deliverable in the contract. There was a change in the appointments at Barcelona,where 12 months of pre-docs were converted into 15 months of post-docs. This is due to the fact that Barcelonaobtained a Marie-Curie site during the network activity, and 12 months of predoctoral position from the networkwould have caused there an oversupply of predoctoral positions (this change had been discussed with the BrusselsOfficer in charge of the network at the midterm review meeting in November 2002 at Paris). Notice also thatone extra month of predoctoral positions has been provided at Paris (team 6) and that two extra months ofpredoctoral positions have been provided at Vienna (team 8).Participant Young predoctoral Young postdoctoral Total (a + b)researchers researchersfinanced by the financed by thenetwork network(person month) (person month)(a)(b)1. UB1 12 (12) 29 (24) 41 (36)2. VUA 0 (0) 28 (24) 28 (24)3. UAB 0 (12) 39 (24) 39 (36)4. UCD 0 (0) 28 (24) 28 (24)5. ULeeds 0 (0) 24 (24) 24 (24)6. UPMC 13 (12) 24 (24) 37 (36)7. NTNU 0 (0) 24 (24) 24 (24)8. TU Vienna 2 (0) 24 (24) 26 (24)9. TAU 0 (0) 24 (24) 24 (24)10. POMI 0 (0) 0 (0) 0 (0)TOTAL 27 (36) 244 (216) 271 (252)This table shows that teams 5, 6, 7, 9 delivered exactly the number of person-months of predoc and postdocemployment indicated in Annex 1 of the contract. Team 3 delivered 15 extra months of post-doc appointmentsbut did not deliver the 12 months of predoc appointments indicated in the contract. Team 8 delivered 2 extramonths of predoc appointments. Team 1 delivered 5 extra months of postdoc appointments and teams 2 and4 delivered 4 extra months of postdoc appointments. Altogether 271 person-months of appointments of youngresearchers were delivered during the four years of network activity, which represents a surplus of 13 monthscompared to the 252 person-months indicated in the Annex 1 of the contract. This possibility to provide extramonths of appointment came from the fact that in some nodes the employment costs turned out to be lowerthan the estimates available during contract negotiations.31

Last periodic report -B.5.3The appointing process (applications sent to the node coordinator and the network coordinator, appointmentsmade by the node after approval by the network coordinator and the panel of coordinators) ensuredthat the young researchers appointed by the network could contribute to the network research objectives andworkplan. This integration was greatly facilitated by the participation of young researchers appointed by thenetwork to the network annual conferences and to the network postdoc workshops which were held duringtheir appointments. As a result the young mathematicians appointed by the network benefited from the helpof senior mathematicians to produce papers on works they started before the appointment and in most casesto start new directions of research. This led to many papers by the young Mathematicians appointed by thenetwork, often in collaboration with a senior member of the host node, see the paper 1 in task 2, papers 14,15, 16, 17 in task 3, paper 1 in task 4, paper 2 in task 6 (section A.2 of the present last periodic report).See also the preprint on sampling by X.Dussau (postdoc in team 3), the preprints Computing the pluricomplexGreen function with two poles and An extremal function for the multiplier algebra of the universal Pick space,by F.Wikstrom, postdoc in team 3, An integral inequality for monotone functions with applications, LinearDifferential equations with solutions in a subspace of the Hardy space and Linear Differential Equations andFunction Spaces of the unit disk, by J.Rattya, postdoc in team 3, and his co-authors, Results on A k (Ω), by O.Lemmers, postdoc in team 3, Regular Dilations in Krein Spaces, by D.Popovici, postdoc in team 8. Notice thatJussi Berhrndt (from T.U.Berlin) has a paper in preparation on operators in indefinite inner product spaces andλ–nonlinear eigenvalue problems. The discussions between E.Strouse and M.Smith about Toeplitz operatorsduring the two months postdoc appointment of M. Smith at Bordeaux and the discussions between A. O’ Farrelland R. Lávička on finely holomorphic functions and the extension of this theory to monogenic functions definedon quaternions during the four months appointment of Lávička at Dublin should also lead to joint papers..Last periodic report -B.5.4During the last year of network activity another young researcher meeting was organized at Paris on January22-23, 2004, see section B.4.2 of the present last periodic report. It seemed very useful to provide formerpostdocs with the possibility of attending network events after the termination of the appointment. Two youngmathematicians (D. Popovici, six months at Bordeaux in 2001-2002 and six months at Vienna in 2002-2003, andM.Smith, two months at Amsterdam and then two months at Bordeaux in 2004), benefited from appointmentsin two different nodes. Notice also that S. Artstein, thesis student in team 9, benefited from two two monthspredoc appointments in Paris in the spring 2002 and in the summer 2004. Training was also organized throughthe series of morning lectures by David-Verdera, Verduyn Lunel-Partington and Vasilevski at the fourth annualconference of the network at Dalfsen, see also section B.4.2 of the present last periodic report.Young Mathematicians appointed by the network were also encouraged to attend conferences external to thenetwork which were relevant to their research plans. For example A. Gustafsson , postdoc in team 4, participatedin a Conference on ”Differential equations and functional equations in the complex domain” in LoughboroughUniversity, 28 June - 1 July, 2003, and in a Function Theory Conference at the London Mathematical Societyon 15 September 2003. Also M.Kopp (postdoc in team 1) and R. Lávička (postdoc in team 4) participated inthe 56th British Mathematical Colloquium, 5-8 April 2004 in Belfast.In all the host nodes, young researchers appointed by the network presented their work in the node’s seminarsand training seminar and had regular discussions with local senior Mathematicians. For example the weeklyseminar on Analysis and Operator Theory played in the Amsterdam node a major role in the training of theyoung researchers C.Mehl and M.Smith. Both gave a series of talks on their ongoing work in the seminar, which32

led to interesting and valuable discussions. Also, the young researchers had regular meetings with their directsupervisors (Kaashoek and Verduyn Lunel for Smith, and Ran for Mehl). Kaashoek and Smith also discussedrecent work of Barachart on the Nehari-Takagi problem in a l p setting. Mehl and Ran made considerableprogress in their joint work with Rodman. A similar policy was implemented in the other host nodes.Last periodic report-B.5.5 Among all the applicants to a predoctoral or postdoctoral position in the networkthere were so far only 5 women. No special measure was taken to promote equal opportunities, but 4 of thesefemale applicants have been appointed (6 months postdoc at Paris for A.Pelczar in 2001-2002, 3 months postdocat Bordeaux for S.Pott in 2003, 2 months predoc at Paris in 2002 and 2 months predoc at Paris in 2003 forS.Artstein, 3 months postdoc at Bordeaux for H. Robinson in 2003). Notice also that 5 of the 13 co-authors ofthe five most significant joint publications listed in part A2 of the final report are women, which gives a ratioof 38,5% of female co-authors.Last periodic report- B.5.6 and B.5.7 Not relevant for the activity of this network.33

Final report- B.3 Training overviewThe vacant positions were publicized throughout the network and advertised on the network homepage, andalso announced through the Banach algebra alias (more than 150 people) run by M.Thomas from California.There were some difficulties in the first year to fill the position at Trondheim, which were circumvented bydelaying the appointment since an excellent candidate was available one year later. There were also somedifficulties to fill two six months positions in Tel-Aviv in 2002, but very good applicants eventually showedup and these two positions were filled during the last year of network activity. There was a slight change inthe balance of pre and post docs, due to the fact that Barcelona got also a Marie Curie site and many predocpositions to offer. So the twelve months of predoc position at Barcelona were changed into postdoc (this wasdiscussed with the Brussels Officer at the midterm review).The appointing process, with applications sent to the node coordinator and the network coordinator, andwith appointments made by the node after approval by the network coordinator and the panel of coordinators,ensured that the young researchers appointed by the network could contribute to the network research objectivesand workplan. This integration was greatly facilitated by the participation of young researchers appointed bythe network to the network annual conferences and to the network postdoc workshops which were held duringtheir appointments. As a result the young mathematicians appointed by the network benefited from the helpof senior mathematicians to produce papers on works they started before the appointment and in most cases tostart new directions of research. In some cases the postdoc or predoc had joint papers with some senior memberof the host node, see the paper 1 in task 2, papers 14, 15, 16, 17 in task 3, paper 1 in task 4, paper 2 in task6 (section A.2 of the last periodic report), the papers [41], [42], [43] of section A2 of the third periodic report,paper [19] of the midterm review report and paper [3] in section A2 of this final report.The main objective of the training program, as stated in Annex 1 of the contract, was the same for thepredoctoral doctoral and senior researchers: to help them to master the interplay between complex and harmonicanalysis, operator theory and the developments of analysis arising from some recent progress in the Geometryof Banach spaces, in order to make significant contributions and, in some cases, decisive breakthroughs in themainstream in this area of Mathematics. The decisive breakthroughs were provided by Tolsa’s work on analyticand continuous analytic capacities, but the whole training program went also along as expected. The number ofjoint papers between young mathematicians appointed by the network and senior mathematicians from the hostnode and of other papers produced by young mathematicians shows the quality of individual training providedwithin the network.An important aspect of collective training in the training programme was the organization of series ofmorning lectures at annual conferences. We give below the program of these lectures, which involved the tenteams of the network and covered all topics of the training programme.St Petersburg (May 2001) Capacities and harmonic approximation, by S.Gardiner and A.O’Farrell (team4)- Linear approximation on Krein spaces and applications, by H.Langer (team 8), Spectral Analysis of selfadjointJacobi matrices, by S. Naboko (team 10)Biarritz (May 2002) The semiadditivity of analytic capacity, by X.Tolsa (team 3), Interpolation of Hardytype spaces, by S.Kislyakov (team 10)- Geometric aspects of approximation in high dimension and connectionsof convex geometry with complexity theory, by V.Milman (team 9), Local theory of operator spaces, by G.Pisier(team 6)Tenerife (May 2003) Bergman function theory, by H. Hedenmalm (team 7)- Control theory for analysts, byN.Nikolski (team 1)-Translation invariant subspaces, by A.Atzmon (Tel-Aviv), and J.Esterle (team 1)Dalfsen (May 2004) Singular integrals and capacities, by G.David (Orsay) and J.Verdera (team 3)-Delayequations and infinite dimensional systems, by J.Partington (team 5) and S.Verduyn Lunel (team 2)- Toeplitzoperators on Bergman spaces, by N.Vasilevski (Mexico).Besides the Conferences, and the more specialized workshops mentioned in Annex 1 of the contract, which34

were very well organized in the second half of network activity after a slow start, it appeared important toorganize a specific meeting where all pre/posdocs appointed by the network would be asked to give a one hourtalk. These three workshops (Bordeaux, January 2002, Paris, November 2002, Paris, January 2004), whichwere very popular among the young mathematicians appointed by the network, are described in section B.2 ofthis final report. Collective and Individual training thus went along as expected, even if it would have beenbetter to be able organize more mutual visits involving predoctoral students not appointed by the network andto exploit Talagrand’s advances on concentration of measure in the training programme. Also direct links withengineering were not developed at network level; the Leiden group (team 2) developed joint work on Chemicalengineering with the Chemistry department of the University of Leiden, and a young pure operator theoristfrom Bordeaux applied abstract operator theoretical tools to problems of image processing in an engineeringlaboratory of Bordeaux university, where he is now working with a permanent research position from CNRS,but these activities remained isolated.Altogether only five women applied to pre/postdoc positions, and four were appointed (three months predocfor Helen Robinson at Bordeaux, four months predoc for Shiri Artstein at Paris, three months postdoc forSandra Pott at Bordeaux, six months postdoc for Anna Pelczar at Paris). This proportion of 6% of personmonths attributed to women is of course not satisfactory. On the other hand the network stimulated jointactivities involving female mathematicians, and five out of the thirteen co-authors of the five papers selected insection A.2 of this final report are women.S.Artstein (Convex Geometry) , from Tel Aviv, twice two months predoc at Paris, B. Klartag (Normedspaces in high dimension), from Tel Aviv, two months predoc at Paris, G. Paouris (Normed spaces in highdimension), from Athens, four months predoc at Paris, T. Matrai (Infinite dimensional differentiability), fromBudapest, three months predoc at Paris, J. Behrndt, from Berlin (indefinite inner product spaces), two monthspredoc at Vienna, and H. Robinson (harmonic analysis), from York, three months predoc at Bordeaux wentback to their home institution after their short term predoc appointment and are now completing their thesis.M. Kopp (Banach and Frechet algebras) obtained his PhD from Cambridge at the end of his 9 months predocposition at Bordeaux, and obtained after that a one year postdoc position at Bordeaux, which ended on May31, 2004. After getting two papers accepted and after looking for positions both in Universities and in privatecorporations he got a job in a bank at London starting June 1, 2004 (his current salary happens to be threetimes larger than the salary of the network coordinator, who was his former tutor from Bordeaux). Notice thatno specific training in financial mathematics was provided at Bordeaux and that this former postdoc obtainedhis extremely well-paid job on the basis of his experience in research in pure mathematics.The following tables describe the postdoc appointments in the network, and in most cases we were able toindicate the current situation of the former postdoc, with unfortunately a reported case in which the formerpostdoc was not able to take advantage of his training in Mathematics (he worked as a translator during thelast academic year). It is too early to evaluate how the former postdocs who had no tenured position beforethe network appointment will benefit from their network experience to find a stable job. For those who alreadyhad a stable job the general success, attested by many papers, of the research performed in the host nodes willcertainly help in a early future these young mathematicians to reach the level of a professorship.35

name country of duration of place of research currentorigin appointment work topic situationPopovici Romania 6 months Bordeaux Operator theory Back to tenure(Timisoara)Kozma Israel 6 months Bordeaux Fourier Analysis Nonpermanentjob (Israel)Kopp Germany 12 months Bordeaux Banach and Frechet algebras Employed at a bankin LondonPott Germany 3months Bordeaux Harmonic Analysis Back to tenure at York(will move to Glasgowon Sept. 1, 2004)Smith England 2 months Bordeaux Harmonic Analysis back to postdoc(York)Olofsson Sweden 24 months Amsterdam Harmonic Analysis nonpermanent job(Stockholm)Smith England 2 months Amsterdam Harmonic Analysis back to postdoc(York)Mehl Germany 2 months Amsterdam Operator theory Associate(Berlin)Marco France 18 months Barcelona Complex Analysis ?Lemmers Netherlands 6 months Barcelona Function Theory ?Dussau France 3 months Barcelona Function theory / one year ATEROperator theory(Bordeaux)Rattya Finland 6 months La Laguna Function Theory lecturer(team 3)(Joensuu)Wikstrom Sweden 6 months La Laguna Function theory Visiting scholar(team 3)Ann Arbor, USAPau Spain 12 months Dublin Potential Analysis lecturer(Barcelona)Gustafsson Sweden 12 months Dublin Function Theory postdoc(Kalmar)Lavicka Czech Rep. 4 months Dublin Function Theory back to tenure(Praha)Michels Germany 12 months Leeds Geometry of associateBanach spaces(Oldenburg)Jaeck France 12 months Lancaster Invariant subspaces/ back to high(team 5) dual algebras school tenureLehner Austria 12 months Paris Operator algebras Postdoc (Graz)Pelczar Poland 6 months Paris Geometry of Back to tenureBanach spaces(Krakow)Bucholz Poland 6 months Besançon Operator algebras ?(team 6)Hartmann Germany 12 months Trondheim Function theory/ back to tenureOperator theory(Bordeaux)Harlouchet France 12 months Trondheim Function theory/ work unrelated toMathematicsBatkai Hungary 12 months Vienna Operator theory back to tenure(Budapest)Lasarow Germany 6 months Vienna Operator theoryPopovici Romania 6 months Vienna Operator theory back to tenure(Timisoara)Oravecz Hungary 12 months Tel Aviv Operator theory back to tenure(Budapest)Brive France 6 months Tel Aviv Entire functions postdoc (Bucarest)Ameur Sweden 6 months Tel Aviv Geometry ofBanach spaces ?36

Final report. B.4 Industry connectionsNot relevant to the activity of this network.Final report- B.5 RecommendationsThe network management was globally very pleased about the relations with the Brussels administrationduring the network activity, including at the midterm meeting on November 20, 2002. It would probably beuseful to introduce some amount of ”evaluation by peers” during the network activity, for example by invitinga scientific expert choosen by the European Commission to assist the Brussels Officer at the midterm reviewmeeting. Also there are problems with currencies different from the euro, which can fluctuate a lot. Since therate of exchange taken into account is the average rate for the first month after the end of the reporting period,nodes with noneuro currencies should be encouraged to complete their appointments, which represent the mainpart of the budget, before the last year of netxwork activity. Other comments are given in the evaluation reportsfrom the network coordinator and the node coordinators.Last periodic report- B.6 DifficultiesThere were technical difficulties with the payments in 2000 and 2001 (several months of delay between thereception by Bordeaux of the initial payment and the first annual payment from Brussels and the reception ofthe due share of these payments by the other nodes). These difficulties have been circumvented for the secondand third periodic payment (delay reduced to one week). Also the number of applicants to postdocs was notvery large (rarely more than 5 or 6) , and there were no applicants at all at Tel-Aviv in 2002 for the two sixmonths positions which were advertized. This problem at Tel-Aviv has been solved, these positions have beenfilled during the last year of network activity, and the overall number of person-months of employment of youngresearchers effectively delivered during network activity is for all the teams at least equal to the number givenin Annex 1 of the contract.37