Abstract
Let T: X → X be a bounded operator on Banach space, whose spectrum σ(T) is included in the closed unit disc \(\overline{\mathbb{D}}\). Assume that the peripheral spectrum \(\sigma(T)\cap\mathbb{T}\) is finite and that T satisfies a resolvent estimate
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 = Lp for some 1 < p < ∞, and T: Lp → Lp is a positive contraction; or (ii) T is polynomially bounded and for all \(\xi\in\sigma(T)\cap\mathbb{T}\), there exists a neighborhood \(\cal{V}\) of ξ such that the set \(\{(\xi-z)(z-T)^{-1}:z\in\cal{V}\cap\overline{\mathbb{D}}^{c}\}\) 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 RittE operator which generalizes the classical notion of Ritt operator. We study these RittE operators and their natural functional calculus.
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Acknowlegement
The authors were supported by the ANR project Non-commutative analysis on groups and quantum groups (No./ANR-19-CE40-0002). Further, the LmB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002)
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Bouabdillah, O., Le Merdy, C. Polygonal functional calculus for operators with finite peripheral spectrum. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2632-y
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DOI: https://doi.org/10.1007/s11856-024-2632-y