Polygonal functional calculus for operators with finite peripheral spectrum

Let T: X → X be a bounded operator on Banach space, whose spectrum σ(T) is included in the closed unit disc \(\overline{\mathbb{D}}\) $$\overline{\mathbb{D}}$$ . Assume that the peripheral spectrum \(\sigma(T)\cap\mathbb{T}\) $$\sigma (T)\cap \mathbb{T}$$ is finite and that T satisfies a resolvent estimate

\(\Vert(z-T)^{-1}\Vert\lesssim\max\{\vert z-\xi\vert^{-1}:\xi\in\sigma(T)\cap\mathbb{T}\},\ \ \ \ z\in\overline{\mathbb{D}}^{c}.\) $$\parallel (z-T{)}^{-1}\parallel \lesssim max\{|z-\xi {|}^{-1}:\xi \in \sigma (T)\cap \mathbb{T}\},z\in {\overline{\mathbb{D}}}^c.$$

We prove that T admits a bounded polygonal functional calculus, that is, an estimate ∥ϕ(T)∥ ≲ sup{∣ϕ(z)∣: z ∈ Δ} for some polygon Δ ⊂ ⅅ and all polynomials ϕ, in each of the following two cases: (i) either X = L^p for some 1 < p < ∞, and T: L^p → L^p is a positive contraction; or (ii) T is polynomially bounded and for all \(\xi\in\sigma(T)\cap\mathbb{T}\) $$\xi \in \sigma (T)\cap \mathbb{T}$$ , there exists a neighborhood \(\cal{V}\) $$\mathcal{V}$$ of ξ such that the set \(\{(\xi-z)(z-T)^{-1}:z\in\cal{V}\cap\overline{\mathbb{D}}^{c}\}\) $$\{(\xi -z)(z-T{)}^{-1}:z\in \mathcal{V}\cap {\overline{\mathbb{D}}}^c\}$$ 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 \(E\subset\mathbb{T}\) $$E\subset \mathbb{T}$$ , of a notion of Ritt_E operator which generalizes the classical notion of Ritt operator. We study these Ritt_E operators and their natural functional calculus.