Unitarily invariant Norms on Operators

Abstract

Let f be a symmetric norm on R n and let B(H) be the set of all bounded linear operators on a Hilbert space H of dimension at least n. Define a norm on B(H) by A f = f (s1(A), . . . , sn(A)), where s k (A) = sup{ A − X : X ∈ B(H) has rank less than k} is the kth singular value of A. Basic properties of the norm • f are obtained including some norm inequalities and characterization of the equality case. Geometric properties of the unit ball of the norm are obtained; the results are used to determine the structure of maps L satisfying L(A) − L(B) f = A − B f for any A, B ∈ B(H).