GROWTH CONDITIONS FOR OPERATORS WITH SMALLEST SPECTRUM

MUSTAFAYEV, H. S.

Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 665-680

Résumé

Let A be an invertible operator on a complex Banach space X. For a given α ≥ 0, we define the class $\mathcal{D}$Aα(Z) (resp. $\mathcal{D}$Aα (Z+)) of all bounded linear operators T on X for which there exists a constant CT>0, such that$\begin{equation*}\Vert A^{n}TA^{-n}\Vert \leq C_{T}\left( 1+\left\vertn\right\vert \right) ^{\alpha },\end{equation*}$for all n ∈ Z (resp. n∈ Z+). We present a complete description of the class $\mathcal{D}$Aα (Z) in the case when the spectrum of A is real or is a singleton. If T ∈ $\mathcal{D}$A(Z) (=$\mathcal{D}$A0(Z)), some estimates for the norm of AT-TA are obtained. Some results for the class $\mathcal{D}$Aα (Z+) are also given.

Détail

DOI : 10.1017/S001708951400055X