Journal of Operator Theory

Volume 53, Issue 2, Spring 2005 pp. 367-380.

Commutants and hyporeflexive closure of operators

Authors: Zhidong Pan
Author institution: Department of Mathematical Sciences, Saginaw Valley State University, University Center, MI 48710, USA

Summary: We show that the commutants of several classes of operators are boundedly reflexive; including Hilbert space triangular operators and Banach space compact cyclic operators, the latter gives an affirmative answer to a question of Don Hadwin and Deguang Han. Under a mild condition on the spectrum of a Banach space operator, we show that the hyporeflexive closure of the operator is boundedly reflexive. With a simpler proof, we obtain a stronger version of a theorem of David Larson and Warren Wogen on algebraic extensions of bitriangular operators. We also show that the commutant of a bitriangular operator on a Hilbert space is reflexive if and only if the bitriangular operator is quasisimilar to a diagonal operator.

Keywords: Commutant, reflexivity, bounded reflexivity, separating vector, cyclic vector.

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