Journal of Operator Theory

Volume 46, Issue 3, Supplementary 2001 pp. 545-560.

On band algebras

Authors: L. Livshits (1), G. MacDonald (2), B. Mathes (3), and H. Radjavi (4)
Author institution: (1) Department of Mathematics, Colby College, Waterville, Maine 04901, USA
(2) Department of Math. and CS, University of Prince Edward Island, Charlottetown, PEI, C1A 4P3, Canada
(3) Department of Mathematics, Colby College, Waterville, Maine 04901, USA
(4) Department of Math., Stats. and CS, Dalhousie University, Halifax, NS, B3H 3J3, Canada

Summary: It is shown that a nest in a Hilbert space $\Fam{H}$ is the lattice of closed invariant subspaces of a band algebra in $\Fam{B}(\Fam{H})$ (i.e. an algebra generated by a semigroup of idempotent operators) if and only if all finite-dimensional atoms of the nest have dimension 1. A canonical operator matrix form for operator bands, developed by the authors, is used to demonstrate that the set of algebraic operators in $\Fam{B}(\Fam{H})$ coincides with the union of all band subalgebras of $\Fam{B}(\Fam{H})$. Several sufficient conditions for an operator band to be reducible and triangularizable are presented, and a new proof is given for a theorem on algebraic triangularizability of arbitrary operator bands.

Keywords: Semigroups, idempotents, invariant subspaces, bands, reducible, irreducible representations

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