The norm of the $k$-th derivative of the $\chi$-symmetric power of an operator

TL;DR

This paper derives exact norms for all order directional derivatives of symmetric tensor powers of finite-dimensional operators and provides bounds for derivatives of immanants, extending prior theoretical results.

Contribution

It presents the exact norms of all order directional derivatives for symmetric tensor powers and bounds for derivatives of immanants, advancing theoretical understanding.

Findings

Exact values for norms of derivatives of symmetric tensor powers

Upper bounds for derivatives of immanants

Extension of previous theoretical results

Abstract

In this paper we present the exact value for the norm of directional derivatives, of all orders, for symmetric tensor powers of operators on finite dimensional vector spaces. Using this result we obtain an upper bound for the norm of all directional derivatives of immanants. This work is inspired in results by R. Bhatia, J. Dias da Silva, P. Grover and T. Jain.