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Journal of Operator Theory

Volume 74, Issue 2, Fall 2015  pp. 281-306.

Growth conditions for conjugation orbits of operators on Banach spaces

Authors:  Heybetkulu Mustafayev
Author institution: Yuzuncu Yil University, Faculty of Sciences, Department of Mathematics, 65080, Van, Turkey

Summary:  Let $A$ be an invertible bounded linear operator on a complex Banach space $X$. With connection to the Deddens algebras, for a given $k\in\mathbb{N}$, we define the class $\mathcal{D}_{A}^{k}$ of all bounded linear operators $T$ on $X$ for which the conjugation orbits $ \{ A^{n}TA^{-n} \}_{n\in \mathbb{Z}}$ satisfies some growth conditions. We present a complete description of the class $\mathcal{D}_{A}^{k}$ in the case when the spectrum of $A$ is positive. Individual versions of Katznelson-Tzafriri theorem and their applications to the Deddens algebras are given. The Hille-Yosida space is used to obtain local quantitative results related to the Katznelson--Tzafriri theorem. Some related problems are also discussed.

Keywords:  operator, Deddens algebra, (local) spectrum, entire function, Katznelson-Tzafriri theorem, Hille-Yosida space

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