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.