The operator-valued parallelism

In this paper, we consider the characterization of norm parallelism problem for trace-class and compact operators on a Hilbert space H  . In particular, for compact operators T,ST,S we show that T∥ST∥S if and only if there exists a unit vector ξ∈Hξ∈H such that |[Tξ,Sξ]|=‖T‖‖S‖. Moreover, for two trace-class operators T,ST,S we prove that T∥ST∥S if and only if there exists a unit λ∈Cλ∈C such that|tr(|T|)+μtr(U⁎S)|≤‖PkerT⁎(T+μS)PkerT‖, where T=U|T|T=U|T| is the polar decomposition of T   and μ=‖T‖‖S‖λ. In addition, we introduce the concept of the absolute value parallelism for bounded linear operators. We show that the relations norm parallelism and the absolute value parallelism are coincident for trace-class operators.