We studied general spectral multiplier theorems for self-adjoint
positive definite operators on $L^2(X, \mu)$, where $X$ is any open
subset of a space of homogeneous type. We developed very general
techniques for showing that the
sharp Hörmander-type spectral multiplier theorems follow from
the appropriate estimates of the $L^2$ norm of the kernel
of spectral multipliers and the Gaussian bounds for the
corresponding heat kernel. The sharp Hörmander-type spectral
multiplier theorems are motivated and connected with sharp
estimates for the critical exponent for the Riesz means
summability, which we also studied. We discussed several
examples, which include sharp spectral multiplier theorems
for a class of scattering operators on $\R^3$ and new
spectral multiplier theorems for Laguerre and Hermite expansions.
The obtained spectral multiplier theorems are widely applicable.