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

ABSTRACT. Let ∨A be the k-th symmetric tensor power of A ∈ Mn(C). In [23], we have expressed the normalized trace of ∨A as an integral of the k-th powers of the numerical values of A over the unit sphere S of C 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 ∨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 Lnorm on the space C(S) 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 -norms on C(S).

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