Spectral-Operator Calculus I: Trace-Form Evaluators and Spectral Growth in the Self-Adjoint Setting

Abstract

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.