Almost everywhere convergence of spectral sums for self-adjoint operators

2021/09/04 by Peng Chen, Chen, Peng, Xuan Thinh Duong +3
Computer Science · Mathematics · #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2109.01778

openalex publication_date 2021/09/04 · openalex created_date 2022/11/30 · openalex updated_date 2026/07/28

Abstract

Let L be a non-negative self-adjoint operator acting on the space L²(X), where X is a metric measure space. Let L=∫₀^∞ λdE _L(λ) be the spectral resolution of L and S_R( L)f=∫₀^R dE _L(λ) f denote the spherical partial sums in terms of the resolution of L. In this article we give a sufficient condition on L such that lim_{R→ ∞} S_R( L)f(x) =f(x), \rm a.e. for any f such that \rm log (2+L) f∈ L²(X). These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrödinger operators with inverse square potentials.

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