English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $T\colon X\to X$ be a bounded operator on Banach space, whose spectrum$\sigma(T)$ is included in the closed unit disc $\overline{\mathbb D}$. Assumethat the peripheral spectrum $\sigma(T)\cap{\mathbb T}$ is finite and that $T$satisfies a resolvent estimate $$\Vert(z-T)^{-1}\Vert\lesssim \max\bigl\{\vertz -\xi\vert^{-1}\, :\,\xi\in \sigma(T)\cap{\mathbb T}\bigr\}, \qquadz\in\overline{{\mathbb D}}^c.$$ We prove that $T$ admits a bounded polygonalfunctional 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 followingtwo 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 ofthese two results extends a theorem of de Laubenfels concerning polygonalfunctional calculus on Hilbert space. Our investigations require theintroduction, for any finite set $E\subset{\mathbb T}$, of a notion of Ritt$_E$operator which generalises the classical notion of Ritt operator. We studythese Ritt$_E$ operators and their natural functional calculus.

Polygonal functional calculus for operators with finite peripheral spectrum