Derivatives of tensor powers and their norms

TL;DR

This paper evaluates the norms of derivatives of tensor power maps, including antisymmetric, symmetric, and permanent maps, providing new formulas and extending previous results in operator theory.

Contribution

It introduces a multilinear version of Russo and Dye's theorem and computes derivatives and their norms for various tensor power maps, including antisymmetric, symmetric, and permanent functions.

Findings

Norms of derivatives of tensor power maps are explicitly computed.

A multilinear version of Russo and Dye's theorem is proved.

Derivatives of the permanent map are evaluated.

Abstract

The norm of the $m$th derivative of the map that takes an operator to its $k$th antisymmetric tensor power is evaluated. The case $m=1$ has been studied earlier by Bhatia and Friedland [R. Bhatia and S. Friedland, Variation of Grassman powers and spectra, Linear Algebra and its Applications, 40:1--18, 1981]. For this purpose a multilinear version of a theorem of Russo and Dye is proved: it is shown that a positive $m$-linear map between $C^{\ast}$-algebras attains its norm at the $m$-tuple $(I, \, I, ..., I).$ Expressions for derivatives of the maps that take an operator to its $k$th tensor power and $k$th symmetric tensor power are also obtained. The norms of these derivatives are computed. Derivatives of the map taking a matrix to its permanent are also evaluated.