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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].

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))
Full Text: DOI
[1] O. Blasco, Vector-valued analytic functions of bounded mean oscillation and geometry of Banach spaces.Illinois J. Math., 41(4):532-558 (1997). · Zbl 0915.46009
[2] F.F. Bonsall. Boundedness of Hankel matrices.J. London Math. Soc., (2), 29:289-300 (1984). · Zbl 0561.47027
[3] F.F. Bonsall. Condition for boundedness of Hankel matrices.Bull. London Math. Soc., 26:171-176 (1994). · Zbl 0822.47027
[4] E.B. Davies.One-Parameter Semigroups. London Math. Society Monographs vol. 15. Academic Press, London, 1980. · Zbl 0457.47030
[5] P. Grabowski and F. M. Callier. Admissible observation operators, semigroup criteria of admissibility.Integ. Equat. Operat. Theory, 25:182-198 (1996). · Zbl 0856.93021
[6] K. Hoffman.Banach Spaces of Analytic Functions. Prentice-Hall, 1962. · Zbl 0117.34001
[7] P. Koosis.Introduction to H p Spaces. Cambridge University Press, Cambridge, 1980. · Zbl 0435.30001
[8] Y. Meyer.Wavelets and operators. Cambridge University Press, Cambridge, 1992. · Zbl 0776.42019
[9] E.W. Packel. A semigroup analogue of Foguel’s counterexample.Proc. Amer. Math. Soc., 21:240-244 (1969). · Zbl 0175.13802
[10] J.R. Partington and G. Weiss. Admissible observation operators for the right shift semigroup.Math. Control Signals Systems, 13:179-192 (2000). · Zbl 0966.93033
[11] A. Simard. Counterexamples concerning powers of sectorial operators on a Hilbert space.Bull. Austral. Math. Soc., 60:459-468 (1999). · Zbl 0945.47035
[12] B. Sz.-Nagy and C. Foia?.Harmonic analysis of Operators on Hilbert Space. North-Holland Publishing Company, Amsterdam, London, 1970.
[13] G. Weiss. Admissible observation operators for linear semigroups.Israel J. Math., 65:17-43 (1989). · Zbl 0696.47040
[14] G. Weiss. Two conjectures on the admissibility of control operators. In F. Kappel W. Desch, editor,Estimation and Control of Distributed Parameter Systems, pages 367-378, Basel, 1991. Birkhäuser Verlag. · Zbl 0763.93041
[15] G. Weiss. A powerful generalization of the Carleson measure theorem. In V. Blondel, E. Sontag, M. Vidyasagar, and J. Willems, editors,Open Problems in Mathematical Systems Theory and Control. Springer Verlag, 1998.
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