Unitarily invariant Norms on Operators

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 $\Vert A{\Vert }_f=f(s_1(A),\dots ,s_n(A))$, where $s_k(A)=\inf \{\Vert A-X\Vert :X\in \mathcal{B}(\mathcal{H})has\; rank\; less\; thank\}$ is the $k$th singular value of $A$. Basic properties of the norm $\Vert \cdot {\Vert }_f$ 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 $\Vert L(A)-L(B){\Vert }_f=\Vert A-B{\Vert }_f$ for any $A,B\in \mathcal{B}(\mathcal{H})$.

12 pages