Polygonal functional calculus for operators with finite peripheral spectrum

arXiv2203.05373MaRDI QIDQ6393338

Oualid Bouabdillah, Christian Le Merdy

Publication date: 10 March 2022

Abstract: Let $TcolonXoX$ be a bounded operator on Banach space, whose spectrum $sigma(T)$ is included in the closed unit disc $overlinemathbbD$. Assume that the peripheral spectrum $sigma(T)capmathbbT$ is finite and that $T$ satisfies a resolvent estimate ${\displaystyle }$Vert(z-T)^{-1}Vertlesssim max�igl{vert z -xivert^{-1}, :,xiin sigma(T)cap{mathbb T}�igr}, qquad zinoverline{{mathbb D}}^c.${\displaystyle }$ We prove that $T$ admits a bounded polygonal functional calculus, that is, an estimate $Vertphi(T)Vertlesssimsupvertphi(z)vert,:,zinDelta$ for some polygon $DeltasubsetmathbbD$ and all polynomials $phi$, in each of the following two cases : (i) either $X=L^p$ for some $1<p<infty$, and $Tcolon{L}^{p}o{L}^p$ is a positive contraction; (ii) or $T$ is polynomially bounded and for all $xiinsigma(T)capmathbbT,$ there exists a neighborhood $mathcalV$ of $xi$ such that the set $(xi-z)(z-T{)}^{-1},:,zinmathcalVcapoverline{mathbbD}^c$ 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 $EsubsetmathbbT$, 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.

Mathematics Subject Classification ID

Functional calculus for linear operators (47A60) Sectorial operators (47B12) Operators on Banach spaces (47B01)

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