Duality between some linear preserver problems. II. Isometries with respect to c-special norms and matrices with fixed singular values

Abstract

Let Fm×n (m≤n) denote the linear space of all m × n complex or real matrices according as F=C or R. Let c=(c1,...,cm)≠0 be such that c1≥⋯≥cm≥0. The c-spectral norm of a matrix Aε{lunate}Fm×n is the quantity {norm of matrix}A{norm of matrix}c ∑ i=I mciσi(A). where σ1(A)≥⋯≥σm(A) are the singular values of A. Let d=(d1,...,dm)≠0, where d1≥⋯≥dm≥0. We consider the linear isometries between the normed spaces (Fm×n,∥·∥c) and (Fm×n,∥·∥d), and prove that they are dual transformations of the linear operators which map L(d) onto L(c), where L(c)= {Xε{lunate}Fm×n:X has singular values c1,...,cm}. © 1988.