New applications to combinatorics and invariant matrix norms of an integral representation of natural powers of the numerical values

Abstract

Let ${\vee }^{k}A$ be the $k$-th symmetric tensor power of $A\in M_n(\mathbb{C})$. In \cite{IAM}, we have expressed the normalized trace of ${\vee }^{k}A$ as an integral of the $k$-th powers of the numerical values of $A$ over the unit sphere ${\mathbb{S}}^n$ of ${\mathbb{C}}^n$ with respect to the normalized Euclidean surface measure $\sigma$. In this paper, we first use this integral representation to construct a family of unitarily invariant norms on $M_n(\mathbb{C})$ and then explore their relations to Schatten-norms of ${\vee }^{k}A$. Another application yields a connection between the analysis of symmetric gauge functions with that of complete symmetric polynomials. Finally, motivated by the work of R. Bhatia and J. Holbrook in \cite{hol}, and as pointed out by R. Bhatia in \cite{bhatia} in the development of the theory of weakly unitarily invariant norms, we provide an explicit form for the weakly unitarily invariant norm corresponding to the $L^4$-norm on the space $C({\mathbb{S}}^n)$ of continuous functions on the sphere. Our result generalize those of R. Bhatia and J. Holbrook in different directions and pave the way to a technique for computing those weakly unitarily invariant norms on $M_n(\mathbb{C})$ that are associated to $L^{2k}$-norms on $C({\mathbb{S}}^n)$.