Unitarily invariant Norms on Operators

TL;DR

This paper introduces a class of unitarily invariant norms on operators derived from symmetric norms on vectors, explores their properties, and characterizes the structure of isometries under these norms.

Contribution

It defines a new family of operator norms based on symmetric vector norms, analyzes their properties, and characterizes the structure of their isometries.

Findings

Established basic properties and inequalities of the norms.

Characterized the structure of isometries preserving these norms.

Analyzed geometric properties of the unit ball of the norms.

Abstract

Let $f$ be a symmetric norm on ${\mathbb R}^n$ and let ${\mathcal B}({\mathcal H})$ be the set of all bounded linear operators on a Hilbert space ${\mathcal H}$ of dimension at least $n$. Define a norm on ${\mathcal B}({\mathcal H})$ by $\|A\|_f = f(s_1(A), \dots, s_n(A))$, where $s_k(A) = \inf\{\|A-X\|: X\in {\mathcal B}({\mathcal H}) \hbox{ has rank less than } k\}$ is the $k$th singular value of $A$. Basic properties of the norm $\|\cdot\|_f$ are obtained including some norm inequalities and characterization of the equality case. Geometric properties of the unit ball of the norm are obtained; the results are used to determine the structure of maps $L$ satisfying $\|L(A)-L(B)\|_f=\|A - B\|_f$ for any $A, B \in {\mathcal B}({\mathcal H})$.