Some inequalities in B(H)

Abstract

Let H denote a separable Hilbert space and let B(H) be the space of bounded and linear operators from H to H. We define a subspace Δ(A,B) of B(H), and prove two inequalities between the distance to Δ(A,B) of each operator T in B(H), and the value sup{‖AnTBn−T‖:n=1,2,…}.