Composition operators that are m-isometries

The results in this paper show that on many Hilbert spaces of analytic functions, including Hardy spaces on the ball and polydisc and standard weighted Bergman spaces on the disk, the only m-isometric composition operators are isometries. Necessary conditions for the symbols of m-isometric composition operators on Dirichlet space are also derived. Finally, it is shown that for symbols with fixed point in the unit disk, the associated composition operator has an m-isometric adjoint if and only if the symbol is an automorphism that fixes the origin.