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

Abstract

Let ∨ k A be the k-th symmetric tensor power of A ∈ Mn(C). In [23], we have expressed the normalized trace of ∨ k A as an integral of the k-th powers of the numerical values of A over the unit sphere S n of C n with respect to the normalized Euclidean surface measure σ. In this paper, we first use this integral representation to construct a family of unitarily invariant norms on Mn(C) and then explore their relations to Schatten-norms of ∨ 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 [10], and as pointed out by R. Bhatia in [6] 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 4norm on the space C(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 Mn(C) that are associated to L 2k -norms on C(S n ).