Certain isometries on $\text{R}$ⁿ

Abstract

Let $\text{R}$ⁿ be the linear space of all real column vectors with n coordinates. Given $\text{x=(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)}{}^{\mathrm{t}}\text{∈}\text{R}{}^{\mathrm{n}}$, its (p, k)-norm is defined by

$$\text{‖x‖}{}_{\mathrm{p,\; k}}\text{= max}\text{(|x}{}_{\mathrm{i}}{}_1\text{|}{}^{\mathrm{p}}\text{+⋯+|x}{}_{\mathrm{i}}{}_{\mathrm{k}}\text{|}{}^{\mathrm{p}}\text{)}{}^{\mathrm{1⧸p}}\text{:1 ⩽ i}{}_1\text{<⋯< i}{}_{\mathrm{k}}\text{⩽ n}$$

where k is a positive integer satisfying 1 ⩽ k ⩽ n and p satisfies 1 ⩽ p ⩽ ∞, whereas its c-norm is defined by

$$\text{|x|}{}_{\mathrm{c}}\text{= max{c}{}^{\mathrm{t}}\text{Px: P is a generalized permutation matrix},}$$

for any nonzero c ∈ $\text{R}$ⁿ. This paper characterizes the isometries for c-norms and (p, k)-norms. Some related results are also considered.