Polygonal functional calculus for operators with finite peripheral spectrum

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.