  Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that
operator $L$ generates the analytic semigroup $\{e^{-tL}\}_{t>0}$ whose kernels
satisfy the standard Gaussian upper bounds. This paper investigates the sharp
weighted inequalities for spectral multipliers $F(L)$ and their commutators
with BMO functions. The obtained results in this paper can be considered to be
improvements of those in \cite{DSY}[Weighted norm inequalities, Gaussian bounds
and sharp spectral multipliers, {\it J. Funct. Anal.}, {\bf 260} (2011),
1106-1131].
