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

TL;DR

This paper introduces new unitarily invariant norms derived from an integral representation of symmetric tensor powers of matrices, linking them to Schatten norms, symmetric functions, and weakly unitarily invariant norms, with applications in matrix analysis.

Contribution

It constructs a family of unitarily invariant norms from an integral representation and explores their relation to Schatten norms and symmetric functions, extending previous work on invariant matrix norms.

Findings

Developed new unitarily invariant norms from integral representations.

Established connections between these norms and Schatten norms.

Provided explicit forms for weakly unitarily invariant norms related to $L^{2k}$-norms.

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 $\vee^kA$ 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 $C(\mathbb{S}^{n})$.