Linear operators preserving unitarily invariant norms of matrices
Chi-Kwong Li, Nam‐Kiu Tsing · Linear and Multilinear Algebra · 1990
Let F m × n 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 F m ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F m×n U ∈ U m (F). and V ∈ Un (F). We characterize those linear operators T F m × n → F m × n which satisfy N (T(A)) = N(A)for all A ∈ F m × 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 F m × n To develop the theory we prove some results concerning unitary operators on F m × n which are of independent interest.