A class of unitarily invariant norms on $B(H)$

Abstract:

Let $H$ be a complex Hilbert space and let $B(H)$ be the algebra of all bounded linear operators on $H$. For $c=(c_{1},\dots ,c_{k})$, where $c_{1}\ge \cdots \ge c_{k}>0$ and $p\ge 1$, define the $(c,p)$-norm of $A\in B(H)$ by \[ \|A\|_{c,p}=\left (\sum _{i=1}^{k} c_{i} s_{i}(A)^{p}\right )^{\frac {1}{p}} , \] where $s_{i}(A)$ denotes the $i$th $s$-numbers of $A$. In this paper we study some basic properties of this norm and give a characterization of the extreme points of its closed unit ball. Using these results, we obtain a description of the corresponding isometric isomorphisms on $B(H)$.