# Journal of Operator Theory

Volume 40, Issue 1, Summer 1998 pp. 35-69.

Reducible semigroups of idempotent operators**Authors**: L. Livshits (1), G. MacDonald (2), B. Mathes (3), and H. Radjavi (4)

**Author institution:**(1) Department of Mathematics and CS, Colby College, Waterville, ME 04901, U.S.A.

(2) Department of Mathematics and CS, University of Prince Edward Island, Charlottetown, PEI, C1A 4P3, Canada

(3) Department of Mathematics and CS, Colby College, Waterville, ME 04901, U.S.A.

(4) Department of Math., Stats and CS, Dalhousie University, Halifax, NS, B3H 3J3, Canada

**Summary:**We study the existence of common invariant subspaces for semigroups of idempotent operators. It is known that in finite dimensions every such semigroup is simultaneously triangularizable. The question of the existence of even one non-trivial invariant subspace is still open in infinite dimensions. Working with semigroups of idempotent operators in Hilbert/Banach vector space settings, we exploit the connection between the purely algebraic structure and the operator structure to show that the answer is affirmative in a number of cases.

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