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 (λ, µ)-commuting operators where λ and µ are real numbers with λµ > 0. Furthermore, we find the restriction of µ for the product of (λ, µ)-commuting quasihyponormal operators to be quasihyponormal. We also give spectral and local spectral relations between λ-commuting operators. Moreover, we show that the operators λ-commuting with a unilateral shift are representable as weighted composition operators.