Polygonal functional calculus for operators with finite peripheral spectrum

Abstract

Let $T:X\to X$ be a bounded operator on Banach space, whose spectrum $\ddot{I}ƒ(T)$ is included in the closed unit disc $\overline{\mathbb{D}}$. Assume that the peripheral spectrum $\ddot{I}ƒ(T)\cap \mathbb{T}$ is finite and that $T$ satisfies a resolvent estimate $$\Vert (z-T{)}^{-1}\Vert \lesssim \max \{|z-\hat{I}¾{|}^{-1}:\hat{I}¾\in \ddot{I}ƒ(T)\cap \mathbb{T}\},z\in {\overline{\mathbb{D}}}^c.$$ We prove that $T$ admits a bounded polygonal functional calculus, that is, an estimate $\Vert \ddot{I}•(T)\Vert \lesssim \sup \{|\ddot{I}•(z)|:z\in \hat{I}”\}$ for some polygon $\hat{I}”\subset \mathbb{D}$ and all polynomials $\ddot{I}•$, in each of the following two cases : (i) either $X=L^p$ for some $1<p<\infty$, and $T:L^p\to L^p$ is a positive contraction; (ii) or $T$ is polynomially bounded and for all $\hat{I}¾\in \ddot{I}ƒ(T)\cap \mathbb{T},$ there exists a neighborhood $\mathcal{V}$ of $\hat{I}¾$ such that the set $\{(\hat{I}¾-z)(z-T{)}^{-1}:z\in \mathcal{V}\cap {\overline{\mathbb{D}}}^c\}$ 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.

Revised version, published in Israël Journal of Mathematics