Generalized Anderson's Inequality

AbstractLet H be a separable infinite-dimensional complex Hilbert space and let B(H) denote the algebra of operators on H into itself. We study the elementary operator Φ:B(H)→B(H) defined by Φ(X)=AXB−CXD, where A and C (respectively, B and D) are nonzero normal commuting operators. We prove that•(i)‖Φ()+‖≥‖‖ for all ∈N(Φ) (the kernel of Φ) and for all ∈B() or•(ii)‖Φ()+‖≥‖‖ for all ∈N(Φ)∩C (the von Neumann–Schatten class), 1≤<∞, ≠2, and for all ∈B() such that Φ()∈C if and only if N()∩N()=N(*)∩N(*)={0}