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

Authors: Jor-Ting Chan, Chi-Kwong Li and Charlies C. N. Tu
Journal: Proc. Amer. Math. Soc. 129 (2001), 1065-1076
MSC (1991): Primary 47D25
Published electronically: October 10, 2000
MathSciNet review: 1814144
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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

\begin{displaymath}\Vert A\Vert _{c,p}=\left (\sum _{i=1}^{k} c_{i} s_{i}(A)^{p}\right )^{\frac{1}{p}} , \end{displaymath}

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 $B(H)$.

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Additional Information

Jor-Ting Chan
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

Chi-Kwong Li
Affiliation: Department of Mathematics, The College of William & Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795

Charlies C. N. Tu
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

Keywords: $s$-number, extreme point, exposed point, isometric isomorphism
Received by editor(s): June 30, 1999
Published electronically: October 10, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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