Subnormal Weighted Shifts on Directed Trees and Composition Operators in L2-Spaces with Nondensely Defined Powers

Abstract

It is shown that for every positive integer n there exists a subnormal weighted shift on a directed tree (with or without root) whose nth power is densely defined while its (n+1)th power is not. As a consequence, for every positive integer n there exists a nonsymmetric subnormal composition operator C in an L2-space over a σ-finite measure space such that Cn is densely defined and Cn+1 is not.