In this talk we consider abstract non-negative self-adjoint operators
on $L^2(X)$ which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition
which in the case of the standard Laplace operator is equivalent to $(p,2)$
restriction estimate of Stein and Tomas. Next we show that in the considered
abstract setting our restriction type condition implies sharp spectral
multipliers and endpoint estimates for the Bochner-Riesz summability.
We also observe that this restriction estimate holds for operators
satisfying dispersive or Strichartz estimates. We describe new spectral
multiplier results for several second order differential operators.
This includes Schr/"odinger operators with inverse square
potentials on $/RR^n$, elliptic operators on compact manifolds and
Schr/"odinger operators on asymptotically conic manifolds.