Polygonal functional calculus for operators with finite peripheral spectrum

TL;DR

This paper establishes that certain operators with finite peripheral spectrum on Banach spaces admit a bounded polygonal functional calculus, extending previous results from Hilbert spaces to broader contexts under specific resolvent and boundedness conditions.

Contribution

It introduces the concept of Ritt$_E$ operators for finite sets on the unit circle and proves polygonal functional calculus results for operators with finite peripheral spectrum.

Findings

Operators with finite peripheral spectrum admit polygonal functional calculus.

Extension of de Laubenfels' theorem from Hilbert spaces to Banach spaces.

Introduction of Ritt$_E$ operators and their functional calculus.

Abstract

Let $T\colon X\to X$ be a bounded operator on Banach space, whose spectrum $\sigma(T)$ is included in the closed unit disc $\overline{\mathbb D}$. Assume that the peripheral spectrum $\sigma(T)\cap{\mathbb T}$ is finite and that $T$ satisfies a resolvent estimate $$\Vert(z-T)^{-1}\Vert\lesssim \max\bigl\{\vert z -\xi\vert^{-1}\, :\,\xi\in \sigma(T)\cap{\mathbb T}\bigr\}, \qquad z\in\overline{{\mathbb D}}^c.$$ We prove that $T$ admits a bounded polygonal functional calculus, that is, an estimate $\Vert\phi(T)\Vert\lesssim \sup\{\vert\phi(z)\vert\, :\, z\in\Delta\}$ for some polygon $\Delta\subset{\mathbb D}$ and all polynomials $\phi$, in each of the following two cases : (i) either $X=L^p$ for some $1<p<\infty$, and $T\colon L^p\to L^p$ is a positive contraction; (ii) or $T$ is polynomially bounded and for all $\xi\in \sigma(T)\cap{\mathbb T},$ there exists a neighborhood $\mathcal V$ of $\xi$ such that the set $\{(\xi-z)(z-T)^{-1}\, :\, z\in{\mathcal V}\cap \overline{{\mathbb D}}^c\}$ is $R$-bounded (here $X$ is arbitrary). Each of these two results extends a theorem of de Laubenfels concerning polygonal functional calculus on Hilbert space. Our investigations require the introduction, for any finite set $E\subset{\mathbb T}$, of a notion of Ritt$_E$ operator which generalises the classical notion of Ritt operator. We study these Ritt$_E$ operators and their natural functional calculus.