Felix Schwenninger, Hans Zwart
Published 2012-07-26Version 1
We show that, given a Banach space and a generator of an exponentially stable $C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any $g$ bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separable Hilbert spaces, we even get admissibility. Furthermore, it is investigated when a bounded calculus can be guaranteed. For this we introduce the new notion of exact observability by direction. Finally, it is shown that the calculus coincides with one for half-plane-operators.
Comments: 30 pages, Extension of the article 'F.L. Schwenninger, H.Zwart, Weakly admissible $\mathcal{H}_{\infty}^{-}$-calculus on reflexive Banach spaces' to be published in Indagationes Mathematicae (2012), DOI:10.1016/j.indag.2012.04.005. Main additions: Generalization to general Banach spaces and relation to the natural half-plane calculus
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