  In this paper we prove spectral multiplier theorems for abstract self-adjoint
operators on spaces of homogeneous type. We have two main objectives. The first
one is to work outside the semigroup context. In contrast to previous works on
this subject, we do not make any assumption on the semigroup. The second
objective is to consider polynomial off-diagonal decay instead of exponential
one. Our approach and results lead to new applications to several operators
such as differential operators, pseudo-differential operators as well as Markov
chains. In our general context we introduce a restriction type estimates \`a la
Stein-Tomas. This allows us to obtain sharp spectral multiplier theorems and
hence sharp Bochner-Riesz summability results. Finally, we consider the random
walk on the integer lattice $\mathbf{Z}^n$ and prove sharp Bochner-Riesz
summability results similar to those known for the standard Laplacian on
$\mathbb{R}^n$.

Analysis of PDEs;Spectral Theory