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The Weiss conjecture on admissibility of observation operators for contraction semigroups. (English) Zbl 1031.93107
Let \(T(t)\) be a \(C_0\)-semigroup on a separable Hilbert space \(H\) with infinitesimal generator \(A\), and let \(C\) be an element of \({\mathcal L}({\mathcal D}(A),{\mathbb C})\). Then \(C\) is called an infinite-time admissible observation operator for \(T(t)\) if there is some \(K>0\) such that \(\|CT(\cdot)x\|_{L^2(0,\infty)}\leq K\|x\|\) for any \(x\in {\mathcal D}(A)\). G. Weiss conjectured [Proc. Int. Conf., Vorau/Austria 1990, ISNM 100, 367-378 (1991; Zbl 0763.93041)] that the following statements are equivalent:
(1) There exists a constant \(M>0\) such that \(\|C(sI-A)^{-1}\|\leq M/\sqrt{\operatorname {Re}s}\), \(s\in {\mathbb C}_+\);
(2) \(C\) is infinite-time admissible.
Weiss proved that (2) implies (1), and also that (1) and (2) are equivalent for normal semigroups and for exponentially stable right-invertible semigroups. J. R. Partington and G. Weiss proved [Math. Control Signals Syst. 13, 179-192 (2000; Zbl 0966.93033)] the Weiss conjecture also for the right-shift semigroup on \(L^2(0,\infty)\). In the paper under review, the Weiss conjecture is proved for contraction semigroups. It is shown that the result contains as particular cases Fefferman’s duality theorem, Bonsall’s theorem on the boundedness of Hankel operators, and the Carleson measure theorem. Let us note also the claim of H. Zwart (see http://perso.uclouvain.be/vincent.blondel/books/openprobs/list.html) that in general the Weiss conjecture is false, as is shown by B. Jacob and H. Zwart in “Disproof of two conjectures of George Weiss” [Memorandum No. 1546, Faculty of Mathematical Sciences, University of Twente, the Netherlands].

MSC:
93C25 Control/observation systems in abstract spaces
93B28 Operator-theoretic methods
47D06 One-parameter semigroups and linear evolution equations
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
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