Higher traces of linear maps on finite-dimensional normed spaces
Authors
Tomasz Kania
Abstract
We prove a unified trace-average formula for the $k$-th higher trace $λ_k(A)=\operatorname{tr}(Λ^k A)$ of a linear operator $A$ on a finite-dimensional normed space. The formula averages the matrix coefficient $w\mapsto\langle(Λ^kA)w, w^*\rangle$ over the unit sphere of $Λ^kX$ against a probability measure $η$; it holds for \emph{all} $A$ if and only if the operator-valued average $T_η=\binom{N}{k}\int w\otimes w^*\operatorname{d}η$ equals the identity. Two natural choices of $η$ satisfy this isotropy: (i) the hypersurface measure when a finite isometry group acts as an orthogonal $2$-design on $Λ^k\mathbb{R}^N$; and (ii) the cone probability measure (no symmetry needed). We also identify a first-order obstruction for hypersurface averages at $k=1$: only degree-$2$ spherical harmonics of the support function contribute.
