Weakly admissible $H^{\infty}(\C_{-})$-calculus on general Banach spaces

TL;DR

This paper develops a new weakly admissible functional calculus for generators of exponentially stable semigroups on Banach spaces, extending the classical $H^{ olinebreak}^ olinebreak ext{infty}$ calculus using Toeplitz operators and system theory concepts.

Contribution

It introduces a weakly admissible calculus for $g(A)$ on Banach spaces and explores conditions for boundedness, including the novel concept of exact observability by direction.

Findings

Constructs a weakly admissible $H^{ olinebreak}^ olinebreak ext{infty}$-calculus using Toeplitz operators.

Establishes admissibility in separable Hilbert spaces.

Identifies conditions for bounded calculus via exact observability by direction.

Abstract

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.