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EMBEDDING OPERATORS INTO C 0 -SEMIGROUPS 6 Proof. If 0 < dim(kerP) < ∞, P is not embeddable by Theorem 3.1. The rest follows from the decomposition H = ker P ⊕ rg P into two invariant subspaces, Lemma 4.6 and embeddability of the identity operator (on any Banach space) into the identity semigroup. □ We finish this section with the following natural question. Open Problem 4.8. Does the necessary condition given in Theorem 3.1 suffice to embed partial isometries on Hilbert spaces? Note that for a partial isometry T the condition is sufficient in each of the following cases: (a) T is injective, i.e., T is an isometry (Theorem 4.4); (b) T is surjective, i.e., T ∗ is an isometry (Corollary 4.5); (c) ker T reduces T. The argument in (c) is similar to the proof of Proposition 4.7 using Theorem 4.4. 5. More examples, a category result In this section we give more examples of embeddable operators. The following result extends the argument from (the proof of) Proposition 4.1 to normal operators. Proposition 5.1. Let T be a normal operator on a Hilbert space. Then T is embeddable if and only if T is injective or dim kerT = ∞. Proof. By the spectral theorem, see e.g. Halmos [13], we may assume that T is a multiplication operator on H = L 2 (Ω, µ) for Ω = σ(T) and some Borel measure µ, say Tf = mf with m ∈ L ∞ (Ω, µ). Note that the essential image of m equals σ(T). Assume first that T is injective, whence µ(m −1 (0)) = 0. Using the principal value of the logarithm on C \ {0}, which is measurable, define (T(t)) t≥0 by T(t)f := e t log m f, f ∈ L 2 (Ω, µ), t ≥ 0. Each T(t) is a bounded operator on L 2 (Ω, µ). Moreover, the family (T(t)) t≥0 is strongly continuous by Lebesgue’s dominated convergence theorem. The semigroup law and the property T(1) = T are clear, so T is embeddable. Assume now dim ker T = ∞. Since T is normal, we have rg T ∗ = (kerT) ⊥ = (kerT ∗ ) ⊥ = rg T. Therefore ker T and rg T reduce T and H = kerT ⊕ rg T. The restriction T | ker T is embeddable by Lemma 4.6. On the other hand, T | rg T is an injective normal operator and hence embeddable by the first part of the proof. This shows that T is embeddable as well. The remaining case is answered in the negative by Theorem 3.1. □ Remark 5.2. The above proof shows that T can be embedded into a normal C 0 -semigroup whenever T is injective. Moreover, if T is invertible, then 0 has positive distance to σ(T) and T can be embedded into a normal C 0 -group with bounded generator employing the same definition for t ∈ R (cf. Proposition 4.1). Remark 5.3. Clearly, also each operator T similar to a normal operator is embeddable if and only if T is injective or dim kerT = ∞. For example, Sz.-Nagy [19] showed that a bijective operator T on a Hilbert space satisfying sup j∈Z ‖T j ‖ < ∞ is similar to a unitary operator and hence embeddable into a C 0 -group by Proposition 4.1. We refer also to van Casteren [2] for
EMBEDDING OPERATORS INTO C 0 -SEMIGROUPS 7 conditions on T to be similar to a self-adjoint operator. We finally refer to Benamara, Nikolski [1] for a characterisation of operators similar to a normal operator as well as for the history of the similarity problem. As a final remark we mention that for separable Hilbert spaces a “typical” contraction is embeddable, see Eisner [7]. Theorem 5.4. The set of all embeddable contractions on a separable infinite-dimensional Hilbert space is residual in the set of all contractions for the weak operator topology. (Recall that a subset of a Baire space is called residual if its complement is of first category.) To prove this fact, one first shows that unitary operators form a residual set, see Eisner [7, Theorem 3.3]. Then Proposition 4.1 finishes the argument. Remark 5.5. This result is an operator-theoretical counterpart to a recent result of de la Rue and de Sam Lasaro [5] from ergodic theory stating that a “typical” measure preserving transformation can be embedded into a measure preserving flow. We refer to Stepin, Eremenko [18] for further results and references. In particular, a “typical” contraction on a separable infinite-dimensional Hilbert space has roots of all order, which is an operator-theoretical analogue of a result of King [16] in ergodic theory. Acknowledgement. The author is deeply grateful to Rainer Nagel for interesting and helpful discussions as well as to the referee whose comments and suggestions improved the paper considerably. References [1] N.-E. Benamara, N. Nikolski, Resolvent tests for similarity to a normal operator, Proc. London Math. Soc. 78 (1999), 585–626. [2] J. A. van Casteren, Operators similar to unitary or selfadjoint ones, Pacific J. Math. 104 (1983), 241–255. [3] J. B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. [4] D. Deckard, C. Pearcy, Another class of invertible operators without square roots, Proc. Amer. Math. Soc. 14 (1963), 445–449. [5] T. de la Rue and J. de Sam Lazaro, The generic transformation can be embedded in a flow (French), Ann. Inst. H. Poincaré, Prob. Statist. 39 (2003), 121–134. [6] N. Dunford and J. T. Schwartz, Linear Operators. I., Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. [7] T. Eisner, A “typical” contraction is unitary, submitted. [8] T. Eisner, B. Farkas, R. Nagel, A. Serény, Weakly and almost weakly stable C 0-semigroups, Int. J. Dyn. Syst. Differ. Equ. 1 (2007), 44–57. [9] K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. [10] M. J. Fisher, The embeddability of an invertible measure, Semigroup Forum 5 (1972/73), 340–353. [11] M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser Verlag, Basel, 2006. [12] M. Haase, Functional calculus for groups and applications to evolution equations, J. Evol. Equ. 7 (2007), 529–554. [13] P. R. Halmos, What does the spectral theorem say?, Amer. Math. Monthly 70 (1963), 241–247. [14] P. R. Halmos, G. Lumer, J. J. Schäffer, Square roots of operators, Proc. Amer. Math. Soc. 4 (1953), 142–149.
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