Journal of Operator Theory

Volume 51, Issue 1, Winter 2004 pp. 71-88.

Local spectral properties of weighted shifts

Authors: T.L. Miller, (1) V.G. Miller, (2) and M.M. Neumann
Author institution: (1) Department of Mathematics and Statistics, Mississippi State University, Drawer MA, MS 39762, USA
(2) Department of Mathematics and Statistics, Mississippi State University, Drawer MA, MS 39762, USA
(3) Department of Mathematics and Statistics, Mississippi State University, Drawer MA, MS 39762, USA

Summary: For a large class of operators on Banach spaces, a natural growth condition is shown to guarantee Bishop's property $(\beta)$. For weighted shifts, this result leads to a sufficient condition in terms of the underlying weight sequence. In the opposite direction, it is shown that every unilateral weighted shift with property $(\beta)$ has fat local spectra and approximate point spectrum a circle, while bilateral weighted shifts with property $(\beta)$ have either fat local spectra or spectrum a circle. A useful new tool is the inner local spectral radius, a counterpart of the standard (outer) local spectral radius. For weighted shifts with Dunford's property {\rm (C)}, both the inner and outer spectral radii turn out to be constant.

Keywords: Weighted shifts, decomposable operators, generalized scalar operators, Bishop's property $(\beta)$

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