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The Weiss conjecture for bounded analytic semigroups. (English) Zbl 1064.47045
The present paper is concerned with the so-called Weiss conjecture on admissible operators for bounded semigroups. Let \(-A\) be the generator of a \(C_0\)-semigroup \((T_t)_{t\geq 0}\) on a Banach space \(X\). A linear bounded operator \(C\) from \(D(-A)\), the domain of \(-A\), to another Banach space is called admissible for \(A\) if there is a positive constant \(M\) such that \(\int_0^\infty \| CT_t x\| ^2dt\leq M^2\| x\| ^2\) for all \(x\in D(-A)\). In G. Weiss [Estimation and control of distributed parameter systems (Vorau, 1990), 367–378 (1991; Zbl 0763.93041)] it was conjectured that \(C\) is admissible for \(A\) if and only if \(\sup_{\text{ Re}\,\lambda <0} \| (-\text{ Re}\,\lambda)^{1/2}C(\lambda-A)^{-1}\| <\infty\). A positive answer to this conjecture is given here for bounded analytic \(C_0\)-semigroups \((T_t)_{t\geq 0}\) with generator \(-A\) if \(A^{1/2}\) is admissible for \(A\).

47D06 One-parameter semigroups and linear evolution equations
93B05 Controllability
47A60 Functional calculus for linear operators
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