# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2014jun14.2059

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

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