On Extended Commuting Operators

Abstract

In this paper, we study properties of extended commuting operators. In particular, we provide the polar decomposition of the product of (lambda, mu)-commuting operators where lambda and mu are real numbers with lambda mu > 0. Furthermore, we find the restriction of mu for the product of (lambda, mu)-commuting quasihyponormal operators to be quasihyponormal. We also give spectral and local spectral relations between lambda-commuting operators. Moreover, we show that the operators lambda-commuting with a unilateral shift are representable as weighted composition operators.