  We investigate $L^p$ boundedness of the maximal Bochner-Riesz means for
self-adjoint operators of elliptic type. Assuming the finite speed of
propagation for the associated wave operator, from the restriction type
estimates we establish the sharp $L^p$ boundedness of the maximal Bochner-Riesz
means for the elliptic operators. As applications, we obtain the sharp $L^p$
maximal bounds for the Schr\"odinger operators on asymptotically conic
manifolds, the harmonic oscillator and its perturbations or elliptic operators
on compact manifolds.
