Remarks on $lambda$-commuting operators (The research of geometric structures in quantum information based on Operator Theory and related topics)

Abstract

In this paper, we study properties of $lambda$-commuting operators. We 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. We also provide the polar decomposition of the product of ($lambda$, $mu$)-commuting operators where $lambda$ and $mu$ are real numbers with $lambda mu$>0 . Finally, we find the restriction of $mu$ for the product of ( $lambda$, $mu$)-commuting quasihyponormal operators to be quasihyponormal.