Spectral-Operator Calculus I: Trace-Form Evaluators and Spectral Growth in the Self-Adjoint Setting
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| Format: | Preprint |
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2025
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| _version_ | 1866909975706599424 |
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| author | Homer, John |
| author_facet | Homer, John |
| contents | We develop Spectral-Operator Calculus (SOC), an axiomatic calculus for scalar evaluation of operator-generated spectral observables. This paper (SOC-I) treats the self-adjoint setting, where observables are bounded Borel transforms and locality is enforced via additivity across spectral partitions. Under unitary invariance, extensivity on orthogonal sums, projector-locality, and a dominated-convergence continuity condition, we prove a rigidity theorem on a natural trace-class envelope: every admissible evaluator agrees with a weighted trace of a single Borel nondecreasing profile applied through the functional calculus. We then introduce a spectral growth taxonomy based on eigenvalue counting asymptotics and show that the polynomial growth regime is stable under the basic constructions of the calculus. These results supply an arithmetic-neutral analytic backbone for subsequent SOC parts and for applications to concrete spectral models. A companion part treats the sectorial/holomorphic setting, where locality is formulated on log-scale via scale-band decompositions and positive kernels rather than spectral projections. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_13721 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral-Operator Calculus I: Trace-Form Evaluators and Spectral Growth in the Self-Adjoint Setting Homer, John Functional Analysis Spectral Theory 47A10, 47B10, 46Lxx We develop Spectral-Operator Calculus (SOC), an axiomatic calculus for scalar evaluation of operator-generated spectral observables. This paper (SOC-I) treats the self-adjoint setting, where observables are bounded Borel transforms and locality is enforced via additivity across spectral partitions. Under unitary invariance, extensivity on orthogonal sums, projector-locality, and a dominated-convergence continuity condition, we prove a rigidity theorem on a natural trace-class envelope: every admissible evaluator agrees with a weighted trace of a single Borel nondecreasing profile applied through the functional calculus. We then introduce a spectral growth taxonomy based on eigenvalue counting asymptotics and show that the polynomial growth regime is stable under the basic constructions of the calculus. These results supply an arithmetic-neutral analytic backbone for subsequent SOC parts and for applications to concrete spectral models. A companion part treats the sectorial/holomorphic setting, where locality is formulated on log-scale via scale-band decompositions and positive kernels rather than spectral projections. |
| title | Spectral-Operator Calculus I: Trace-Form Evaluators and Spectral Growth in the Self-Adjoint Setting |
| topic | Functional Analysis Spectral Theory 47A10, 47B10, 46Lxx |
| url | https://arxiv.org/abs/2512.13721 |