The trace as an average over the unit sphere of a normed space with a 1-symmetric basis

TL;DR

This paper extends the classical trace formula to finite-dimensional normed spaces with a 1-symmetric basis, broadening its applicability beyond Euclidean spaces, including -normed spaces.

Contribution

It generalizes the trace integral formula to spaces with a 1-symmetric basis, including -norms, which was not previously known.

Findings

Derived a new trace formula for spaces with 1-symmetric basis

Extended the classical Euclidean trace formula to -normed spaces

The result applies to -norms in ^N for the first time.

Abstract

We generalise the formula expressing the matrix trace of a given square matrix as the integral of the numerical values of $A$ over the Euclidean sphere to the unit spheres of finite-dimensional normed spaces that have a 1-symmetric basis. Our result is new even in the case of $\ell_p$-norms in $\mathbb{R}^N$ for $p\neq 2$.