Isometric and Quasi-isometric weighted composition operators

In this paper we characterize $m$-isometric and quasi-$m$-isometric weighted composition operators on the Hilbert space $L^2(\mu )$. Also, we find that normal-$m$-isometry and normal quasi-$m$-isometry weighted composition operators have finite spectrum. Consequently we have the results for composition and multiplication operators. In addition, we prove that for $m\ge 2$, a multiplication operator is $m$-isometry (quasi-$m$-isometry) if and only if it is $2$-isometry (quasi-$2$-isometry). Some examples are provided to illustrate our results.