  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.
