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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.16501 |
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| _version_ | 1866912659250610176 |
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| author | Kania, Tomasz |
| author_facet | Kania, Tomasz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16501 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher traces of linear maps on finite-dimensional normed spaces Kania, Tomasz Functional Analysis Differential Geometry Primary 15A75, 52A40, Secondary 15A15, 46B20, 42C10, 20C15, 52A20 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. |
| title | Higher traces of linear maps on finite-dimensional normed spaces |
| topic | Functional Analysis Differential Geometry Primary 15A75, 52A40, Secondary 15A15, 46B20, 42C10, 20C15, 52A20 |
| url | https://arxiv.org/abs/2510.16501 |