Unitarily invariant Norms on Operators

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 $A, B \in {\mathcal B}({\mathcal H})$.