Polygonal functional calculus for operators with finite peripheral spectrum
Oualid Bouabdillah, Christian Le Merdy
TL;DR
The paper extends polygonal functional calculus from Hilbert spaces to Banach spaces for operators with finite peripheral spectrum, under a resolvent bound $\|R(z,T)\| \lesssim \max_{\xi\in\sigma(T)\cap\mathbb T} |z-\xi|^{-1}$, by introducing and exploiting the concept of Ritt$_E$ operators. It develops a detailed correspondence between Ritt$_E$ properties and $H^\infty(E_s)$ functional calculus, via a decomposition of unity and $R$-boundedness techniques, to obtain bounded polygonal calculi from polynomial boundedness. The results yield two main Banach-space extensions of de Laubenfels’ Hilbert-space theorem: (i) for $X=L^p$ with a positive contraction (or contractively regular) and finite peripheral spectrum, and (ii) for general Banach spaces under $R$-boundedness hypotheses near the peripheral spectrum, linking resolvent behavior to polygonal calculi. Collectively, these findings advance the understanding of spectral-analytic inequalities for non-Hilbertian settings and have implications for stability and functional calculus in Banach spaces, including $L^p$-spaces.
Abstract
Let $T\colon X\to X$ be a bounded operator on Banach space, whose spectrum $σ(T)$ is included in the closed unit disc $\overline{\mathbb D}$. Assume that the peripheral spectrum $σ(T)\cap{\mathbb T}$ is finite and that $T$ satisfies a resolvent estimate $$\Vert(z-T)^{-1}\Vert\lesssim \max\bigl\{\vert z -ξ\vert^{-1}\, :\,ξ\in σ(T)\cap{\mathbb T}\bigr\}, \qquad z\in\overline{\mathbb D}^c.$$ We prove that $T$ admits a bounded polygonal functional calculus, that is, an estimate $\Vertφ(T)\Vert\lesssim \sup\{\vertφ(z)\vert\, :\, z\inΔ\}$ for some polygon $Δ\subset{\mathbb D}$ and all polynomials $φ$, 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 $ξ\in σ(T)\cap{\mathbb T},$ there exists a neighborhood $\mathcal V$ of $ξ$ such that the set $\{(ξ-z)(z-T)^{-1}\, :\, z\in{\mathcal V}\cap \overline{\mathbb D}^c\}$ 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.