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Analytic Besov functional calculus for several commuting operators

Charles Batty, Alexander Gomilko, Dominik Kobos, Yuri Tomilov

Abstract

This paper investigates when analytic Besov functions of $n$ variables act on the generators of $n$ commuting $C_0$-semigroups on a Banach space. The theory for $n=1$ has already been published, and the present paper uses a different approach to that case as well as extending to the cases when $n\ge2$. It also clarifies some spectral mapping properties and provides some operator norm estimates.

Analytic Besov functional calculus for several commuting operators

Abstract

This paper investigates when analytic Besov functions of variables act on the generators of commuting -semigroups on a Banach space. The theory for has already been published, and the present paper uses a different approach to that case as well as extending to the cases when . It also clarifies some spectral mapping properties and provides some operator norm estimates.
Paper Structure (9 sections, 31 theorems, 161 equations)

This paper contains 9 sections, 31 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. The algebra $\mathcal{B}^n$
  3. Reproducing formulas and shifts
  4. The $\mathcal{B}^n$-calculus: set-up
  5. The $\mathcal{B}^n$-calculus: developments
  6. Possible simplifications
  7. Spectral mapping properties
  8. Estimates for functions and operators
  9. Appendix

Key Result

Proposition 2.1

Let $f$ be a holomorphic function on ${\mathbb C}_+^n$, and assume that $\|f\|_{\mathcal{B}^n_\Omega}$ is finite for every singleton subset $\Omega$ of $I_n$. Then $f$ is bounded and uniformly continuous on ${\mathbb C}_+^n$.

Theorems & Definitions (61)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.6
  • proof
  • ...and 51 more