Certain isometries on [formula omitted

Abstract

Let R n be the linear space of all real column vectors with n coordinates. Given x=(x 1,…,x n) t ∈ R n , its ( p, k)-norm is defined by ‖x‖ p, k = max (|x i 1| p+⋯+|x i k| p) 1⧸p:1 ⩽ i 1 <⋯< i k ⩽ n where k is a positive integer satisfying 1 ⩽ k ⩽ n and p satisfies 1 ⩽ p ⩽ ∞, whereas its c-norm is defined by |x| c = max{c tPx: P is a generalized permutation matrix}, for any nonzero c ∈ R n . This paper characterizes the isometries for c-norms and ( p, k)-norms. Some related results are also considered.