Duality between some linear preserver problems. III: c-spectral norms and (skew)-symmetric matrices with fixed singular values
This paper is a continuation of the authors' earlier papers [part I: Linear Multilinear Algebra 23, No.4, 353-362 (1988; Zbl 0668.15014) and part II: Linear Algebra Appl. 110, 181-212 (1988; Zbl 0655.15026)]. Here they consider the problem of a linear preserver on the matrix space V of all \(n\times n\) symmetric and skew-symmetric matrices over the complex field C or the real field R. They prove that the linear operator T: \(V\to V\) satisfies \(\| T(A)=_ d=\| A\|_ c\) for all A in V if and only if \(T^*(\Sigma (d))=\Sigma (c)\). Necessary and sufficient conditions for the existence of such operators and their forms are obtained. In the process of proving this, they obtain a characterization of extreme points of the unit ball of V with respect to \(\| \cdot \|_ c\).
scientific article; zbMATH DE number 4171134
scientific article