Functional calculus under the Tadmor--Ritt condition, and free interpolation by polynomials of a given degree
Let \(T: X \to X\) be a bounded operator in a Banach space \(X\). Suppose that \(T\) satisfies the Tadmor--Ritt condition (TR): \(\| (zI-T)^{-1} \| \leq C | z-1 |^{-1}\) for all \(| z| >1\). The problem under consideration is to estimate the rate of growth of the following constant: \(C_T (n)=\sup \{ \| p(T)\|: \text{ {deg}}(p)\leq n, \| p \|_{\infty} \leq 1\}\). The author proves that \(C_T(n) \leq (C+1) \log (e^2 n), n \geq 1\). Moreover, there exists an operator \(T\) satisfying (TR) with a constant \(C \leq \pi/2+1\) and such that \(C_T(n) \geq 1/2540 \log(e^2 n)\) for all \(n \geq 1\).
Carleson type free interpolation by polynomials
scientific article; zbMATH DE number 2091738
scientific article