  Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$,
where $X$ is a metric measure space. Let ${ L}=\int_0^{\infty} \lambda dE_{
L}({\lambda})$ be the spectral resolution of ${ L}$ and $S_R({ L})f=\int_0^R
dE_{ L}(\lambda) f$ denote the spherical partial sums in terms of the
resolution of ${ L}$. In this article we give a sufficient condition on $L$
such that $$ \lim_{R\rightarrow \infty} S_R({ L})f(x) =f(x),\ \ {\rm a.e.} $$
for any $f$ such that ${\rm log } (2+L) f\in L^2(X)$.
  These results are applicable to large classes of operators including
Dirichlet operators on smooth bounded domains, the Hermite operator and
Schr\"odinger operators with inverse square potentials.
