m-Isometric Composition Operators on Discrete Spaces

Abstract In this paper we study composition operators on discrete spaces $$ L^{2}(\mu ) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We establish the classification of underlying graphs of such operators. We obtain the characterization of m -isometric composition operators such that the underlying graph has one cycle. The paper contains also the solution of the Cauchy dual subnormality problem in the class of such composition operators.