On extended commuting operators

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