Higher traces of linear maps on finite-dimensional normed spaces
Tomasz Kania
TL;DR
The paper studies how higher traces $\lambda_k(A)=\operatorname{tr}(\Lambda^kA)$ can be realized as averages of matrix coefficients over unit spheres in the exterior power space. It introduces an isotropy condition $T_\eta=I_V$ (with $T_\eta=n_k\int_{S_V} w\otimes w^*\,d\eta(w)$) that exactly characterizes when the trace-average formula holds for all linear maps $A$, and identifies two robust isotropy mechanisms: cone measures (always isotropic) and hypersurface measures under finite-group orthogonal $2$-design symmetry. The work also provides explicit counterexamples illustrating limitations of the hypersurface measure, develops a quantitative anisotropy tensor $\mathcal{A}_X$ linking trace deviations to geometric anisotropy, and analyzes $\alpha$-cone measures, showing isotropy at $\alpha=1$ for convex bodies and a first-order obstruction for $\alpha\neq1$ tied to degree-2 spherical harmonics. Overall, it unifies and extends classical trace-average results to arbitrary norms and exterior powers, with clear geometric and group-theoretic criteria for when the averages reproduce the exact traces.
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.