Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

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544 A. Olofsson / Journal of Functional Analysis 236 (2006) 517–545 Proof. Recall the form of an invariant subspace calculated in Theorem 5.1. By Corollary 4.1 the characteristic operator function WT is an isometric multiplier from the Hardy space A1(DT ∗) into the Hardy space A1(DT ) with range equal to I1,T . This completes the proof of the corollary. ✷ The previous results yield the following consequence concerning the index dim EI of a shift invariant I of An(E). Corollary 5.2. Let I be a shift invariant subspace of An(E) with E separable. Set H = An(E)⊖I and T = S ∗ n |H. Then dim Dn,T dim E. If the defect index dim Dn,T is finite, then dim EI = dim E − dim Dn,T + dim D ∗ n,T . Proof. The first inequality dim Dn,T dim E is evident by the fact that the operator ˆVn ∈ L(Dn,T , E) is an isometry (see [22, Section 7]). The second inequality is evident by the description of the wandering subspace EI in Theorem 5.2. ✷ We notice also that dim EI = dim D ∗ n,T if ˆVn(Dn,T ) = E. It has been known for some time that even in the scalar case E = C the index dim EI of a shift invariant subspace I of An(C) for n 2 can equal any positive integer or +∞. This was first proved by Apostol et al. [5] using dual algebras, and later more explicit constructions have been found by Hedenmalm et al. [19] and others. In the context of the Hardy space A1(E) with E separable it is a result of Halmos [12] that the index dim EI of a shift invariant subspace I of A1(E) cannot exceed the index of the whole space A1(E) meaning that dim EI dim E. This inequality is naturally interpreted as an inequality of defect indexes as follows. Proposition 5.1. Let T ∈ L(H) be a contraction operator in the class C0· acting on a separable Hilbert space H. Then dim DT ∗ dim DT . Proof. It is known that the characteristic operator function WT has non-tangential boundary values WT (e iθ ) in the strong operator topology for a.e. e iθ ∈ T. A well-known argument then shows that the operator WT (e iθ ) is an isometry in L(DT ∗, DT ) for a.e. e iθ ∈ T (see [26, Chapter V]). This gives the conclusion of the proposition. ✷ For an n-hypercontraction T ∈ L(H) in the class C0· and n 2 the corresponding inequality dim D ∗ n,T dim Dn,T of defect indexes is not true in general for the reasons quoted above. References [1] J. Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982) 608–631. [2] J. Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985) 203–217. [3] A. Aleman, S. Richter, C. Sundberg, Beurling’s theorem for the Bergman space, Acta Math. 177 (1996) 275–310. [4] C.-G. Ambrozie, M. Engliš, V. Müller, Operator tuples and analytic models over general domains in C n ,J.Operator Theory 47 (2002) 287–302. [5] C. Apostol, H. Bercovici, C. Foias, C. Pearcy, Invariant subspaces, dilation theory and the structure of the predual of a dual algebra. I, J. Funct. Anal. 63 (1985) 369–404. [6] J. Arazy, M. Engliš, Analytic models for commuting operator tuples on bounded symmetric domains, Trans. Amer. Math. Soc. 355 (2003) 837–864.
A. Olofsson / Journal of Functional Analysis 236 (2006) 517–545 545 [7] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950) 337–404. [8] C. Badea, G. Cassier, Constrained von Neumann inequalities, Adv. Math. 166 (2002) 260–297. [9] J.A. Ball, N. Cohen, de Branges–Rovnyak operator models and systems theory: A survey, in: Topics in Matrix and Operator Theory, Rotterdam, 1989, in: Oper. Theory Adv. Appl., vol. 50, Birkhäuser, 1991, pp. 93–136. [10] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, vol. II, Oper. Theory Adv. Appl., vol. 63, Birkhäuser, 1993. [11] P.R. Halmos, Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950) 125–134. [12] P.R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961) 102–112. [13] H. Hedenmalm, A factorization theorem for square area-integrable analytic functions, J. Reine Angew. Math. 422 (1991) 45–68. [14] H. Hedenmalm, A factoring theorem for a weighted Bergman space, Algebra i Analiz 4 (1992) 167–176; St. Petersburg Math. J. 4 (1993) 163–174. [15] H. Hedenmalm, A factoring theorem for the Bergman space, Bull. London Math. Soc. 26 (1994) 113–126. [16] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Springer, 2000. [17] H. Hedenmalm, Y. Perdomo, Mean value surfaces with prescribed curvature form, J. Math. Pures Appl. 83 (2004) 1075–1107. [18] H. Hedenmalm, K. Zhu, On the failure of optimal factorization for certain weighted Bergman spaces, Complex Variables Theory Appl. 19 (1992) 165–176. [19] H. Hedenmalm, S. Richter, K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. Reine Angew. Math. 477 (1996) 13–30. [20] A. Olofsson, A monotonicity estimate of the biharmonic Green function, Arch. Math. (Basel) 82 (2004) 240–244. [21] A. Olofsson, Wandering subspace theorems, Integral Equations Operator Theory 51 (2005) 395–409. [22] A. Olofsson, An operator-valued Berezin transform and the class of n-hypercontractions, submitted for publication. [23] S. Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386 (1988) 205–220. [24] S.M. Shimorin, Wold-type decompositions and wandering subspaces of operators close to isometries, J. Reine Angew. Math. 531 (2001) 147–189. [25] S.M. Shimorin, On Beurling-type theorems in weighted l 2 and Bergman spaces, Proc. Amer. Math. Soc. 131 (2003) 1777–1787. [26] B. Sz.-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, 1970. [27] K. Zhu, Sub-Bergman Hilbert spaces on the unit disk, Indiana Univ. Math. J. 45 (1996) 165–176.
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