Almost everywhere convergence of spectral sums for self-adjoint operators

Abstract

Let L L be a non-negative self-adjoint operator acting on the space L 2 ( X ) L^2(X) , where X X is a positive measure space. Let L = ∫ 0 ∞ λ d E L ( λ ) { L}=\int _0^{\infty } \lambda dE_{ L}({\lambda }) be the spectral resolution of L L and S R ( L ) f = ∫ 0 R d E L ( λ ) f S_R({ L})f=\int _0^R dE_{ L}(\lambda ) f denote the spherical partial sums in terms of the resolution of L { L} . In this article we give a sufficient condition on L L such that lim R → ∞ S R ( L ) f ( x ) = f ( x ) , a.e. \begin{equation*} \lim _{R\rightarrow \infty } S_R({ L})f(x) =f(x),\ \ \text {a.e.} \end{equation*} for any f f such that log ⁡ ( 2 + L ) f ∈ L 2 ( X ) \operatorname {log}(2+L) f\in L^2(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.