Examples of $m$-isometric tuples of operators on a Hilbert space

The $m$-isometry of a single operator in Agler and Stankus \cite{AS} was naturally generalized to the $m$-isometric tuple of several commuting operators by Gleason and Richter \cite{GR}. Some examples of $m$-isometric tuples including the recently much studied Arveson-Drury $d$-shift were given in \cite{GR}. We provide more examples of $m$-isometric tuples of operators by using sums of operators or products of operators or functions of operators. A class of $m$-isometric tuples of unilateral weighted shifts parametrized by polynomials are also constructed. The examples in Gleason and Richter \cite{GR} are then obtained by choosing some specific polynomials. This work extends partially results obtained in several recent papers on the $m$-isometry of a single operator.