Linear Operators Preserving Unitarily Invariant Norms of Matrices

Abstract

Let Fm×x be the set of all m × n matrices over the field F(= C or R). Denote by Un(F) the group of all n × n unitary or orthogonal matrices according as F = C or F = R. A norm N(·) on Fm×x, is unitarily invariant if N(UAV) = N(A) for all A ∊ Fm×n, U ∊ Um(F), and V ∊ Un(F). We characterize those linear operators T: Fm×x → Fm×n, which satisfy N(T(A)) = N(A) for all A ∊ Fm×n for a given unitarily invariant norm N(·). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in Fm×n. To develop the theory we prove some results concerning unitary operators on Fm×n, which are of independent interest. © 1990, Taylor & Francis Group, LLC. All rights reserved.