Tensor-Lifted Multivariate Functional Calculus Beyond Commutativity and Boundedness

TL;DR

This paper introduces a comprehensive multivariate functional calculus that captures spectral and algebraic structures of operators, including non-commuting, unbounded, and non-self-adjoint cases, with explicit convergence guarantees.

Contribution

It develops the first unified framework incorporating nilpotent structures, tensor lifting for non-commuting operators, and convergence theory for a broad class of operators.

Findings

Includes nilpotent derivatives in the calculus for generalized eigenspaces.

Treats non-commuting operators via tensor lifting into commuting systems.

Establishes a two-level convergence theory with explicit error bounds.

Abstract

Classical functional calculus is primarily spectral, capturing eigenvalue information through resolvent methods while largely ignoring nilpotent structure. Building on the projector-nilpotent characterization developed in our companion work, we introduce a multivariate functional calculus for arbitrary operators that incorporates both spectral and algebraic information. The framework has three main components. First, nilpotent derivative terms are explicitly included in the functional expansion, allowing the calculus to capture generalized eigenspaces and Jordan structures beyond classical resolvent methods. Second, tensor lifting treats non-commuting operators by embedding them into a commuting system on a tensor-product space. Third, a two-level convergence theory is established: Level 1 proves existence through strong resolvent convergence implying strong operator topology convergence, while Level 2 provides stability through norm resolvent convergence implying operator norm convergence with explicit error bounds. The framework simultaneously handles discrete, continuous, and hybrid spectra. Results are developed for bounded operators, unbounded self-adjoint operators recovering the classical spectral theorem, and unbounded non-self-adjoint operators with compact resolvent. For operators without compact resolvent, we introduce a compactifying regularization method based on perturbation by a positive self-adjoint operator with compact resolvent. The proposed framework is compatible with existing functional calculi and recovers their behavior under corresponding assumptions. To our knowledge, this is the first unified framework simultaneously addressing non-commutativity, non-self-adjointness, and unboundedness while explicitly preserving nilpotent structure and providing convergence guarantees.