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

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 AϵFm×n is the quantity ‖A‖c∑i=Imciσi(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)= {XϵFm×n:X has singular values c1,…,cm}.