Journal of Operator Theory

Volume 52, Issue 1, Summer 2004 pp. 89-101.

Peripheral point spectrum and growth of powers of operators

Authors: Omar el-Fallah (1) and Thomas Ransford (2)
Author institution: (1) Departement de mathematiques et informatiques, Universite Mohammed V, Avenue Ibn Battouta, BP 1014, Rabat, Morocco
(2) Departement de mathematiques et informatiques, Universite Laval, Quebec (QC), G1K 7P4, Canada

Summary: Let $E$ be a closed subset of the unit circle. A result of Nikolski shows that, if $T$ is an operator on a separable Hilbert space whose point spectrum contains $E$, and if $0<\alpha<\dim_{\rm H}E$ (the Hausdorff dimension of $E$), then $\sum\limits_n n^{\alpha-1}\|T^n\|^{-2}<\infty$. We complement this result by showing that, for each $\beta>\overline{\dim}_{\rm B}E$ (the upper box dimension of $E$), there exists an operator $T$ on a separable Hilbert space, whose point spectrum contains $E$, and such that $\sum\limits_n n^{\beta-1}\|T^n\|^{-2}=\infty$. We also prove some more refined results along the same lines.

Keywords: Point spectrum, powers of operators, resolvent

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