Functional calculus for semigroup generators via transference

Abstract

In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if $-A$ generates a $C_0$-semigroup on a Hilbert space, then for each $\tau >0$ the operator $A$ has a bounded calculus for the closed ideal of bounded holomorphic functions on a (sufficiently large) right half-plane that satisfy $f(z)=O(e^{-\tau \text{Re}(z)})$ as $|z|\to \infty$. The bound of this calculus grows at most logarithmically as $\tau \searrow 0$. As a consequence, $f(A)$ is a bounded operator for each holomorphic function $f$ (on a right half-plane) with polynomial decay at $\infty$. Then we show that each semigroup generator has a so-called (strong) $m$-bounded calculus for all $m\in \mathbb{N}$, and that this property characterizes semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called $\gamma$-bounded semigroups, the Hilbert space results actually hold in general Banach spaces.

25 pages, updated version. Final version published in Journal of Functional Analysis

References in corpus (1)

• The $H^{\infty }$-Functional Calculus and Square Function Estimates

Cited by in corpus (5)

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• On measuring unboundedness of the $H^{\infty }$-calculus for generators of analytic semigroups
• Functional calculus for $C_0$-groups using (co)type
• Functional calculus on real interpolation spaces for generators of $C_0$-groups
• A Besov algebra calculus for generators of operator semigroups and related norm-estimates