Journal of Operator Theory

Volume 59, Issue 1, Winter 2008 pp. 193-210.

Spectral pictures of operator-valued weighted bi-shifts

Authors: Abdellatif Bourhim
Author institution: Departement de mathematiques et de statistique, Universite Laval, Quebec (Quebec), Canada G1K 7P4

Summary: In this paper, we introduce operator-valued weighted bi-shifts on the Hilbert space $l^2(\mathbb{N},\mathcal{H})$, of all square-summable sequences whose elements are in a complex Hilbert space $\mathcal{H}$, and study their spectral and local spectral properties. We determine the spectrum and its parts of such bi-shifts, and compute their local spectrum at most points of $l^2(\mathbb{N},\mathcal{H})$. Furthermore, we provide necessary and sufficient conditions for an operator-valued weighted bi-shift to enjoy the single-valued extension property.

Keywords: Weighted shift, weighted bi-shift, spectrum, point spectrum, approximate point spectrum, local spectrum, the single-valued extension property.

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