Self adjoint strictly cyclic operator algebras

Mary Rodriguez Embry

Vol. 52 (1974), No. 1, 53–57

DOI: 10.2140/pjm.1974.52.53

Abstract

A strictly cyclic operator algebra 𝒜 on a Hilbert space X is a uniformly closed subalgebra of Z(X) such that 𝒜x₀ = X for some x₀ in X. In this paper it is shown that if 𝒜 is a strictly cyclic self-adjoint algebra, then (i) there exists a finite orthogonal decomposition of X,X = ∑ _J=1ⁿ ⊕ M_j, such that each M_j reduces 𝒜 and the restriction of 𝒜 to M_j is strongly dense in Z(M_j) and (ii) the commutant of 𝒜 is finite dimensional.

Mathematical Subject Classification 2000

Primary: 46L05

Milestones

Received: 6 October 1973

Revised: 30 January 1974

Published: 1 May 1974