Duality between some linear preserver problems. III. c-spectral norms and (skew)-symmetric matrices with fixed singular values

AbstractLet F denote either the complex field C or the real field R. Let V be Sn(F) or Kn(F), the vector spaces of all n × n symmetric and skew-symmetric matrices, respectively, over F. For c=(c1,…,cn)≠0 with c1⩾ ⋯ ⩾cn⩾0, the c-spectral norm of a matrix A∈V is the quantity ‖A‖c = ∑i=lnciσi(A), where σ1(A)⩾ ⋯ ⩾σn(A) are the singular values of A. Let d=(d1,…,dn)≠0 with d1⩾ ⋯ ⩾dn⩾0. We study the linear isometries between the normed spaces (V,‖·‖c) and (V,‖·‖d), by using the fact that they are dual transformations of the linear operators which map ∑(d) onto ∑(c), where ∑(c) = {X∈V:X has singular values c1,…,cn}. It is shown that such isometries (and hence their dual transformations) exist if and only if c and d are scalar multiples of each other. In such case, we completely determine the structure of such isometries, and prove that they and their dual transformations belong to a same class of operators. In the proof, we obtain characterizations of the extreme points of the unit ball in V (for different cases) with respect to ‖·‖c, which is of independent interest