Isometries for the vector (p,q) norm and the induced (p,q) norm

Let M m,n be the linear space of m×n matrices over , where For 1≤p≤∞, let be the -norm of the column vector x. Suppose 1≤p,q≤∞. Define the vector (p,q) norm on M m,n by where Aj is the jth column of A for j = 1, …n, and define the induced(p,q) norm on Mm,n by These definitions include the maximum row sum norm. maximum column sum norm, and operator norm, etc as special cases. In this paper, we characterize the linear isometrics for the vector (p,q) norm and the induced (p,q) norm on M m,n . The proofs depend on certain inequalities involving these norms and the Frobenius norm. These inequalities are of independent interest.