Subnormal Weighted Shifts on Directed Trees and Composition Operators in $L^2$>-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 L ²-space over a σ-finite measure space such that Cⁿ is densely defined and $C^{n+1}$ is not.