Weakly admissible $H^{\infty}(\C_{-})$-calculus on general Banach spaces
Abstract
We show that, given a Banach space and a generator of an exponentially stable -semigroup, a weakly admissible operator can be defined for any 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.
Cite
@article{arxiv.1207.6229,
title = {Weakly admissible $H^{\infty}(\C_{-})$-calculus on general Banach spaces},
author = {Felix Schwenninger and Hans Zwart},
journal= {arXiv preprint arXiv:1207.6229},
year = {2012}
}
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