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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.

Subnormal Weighted Shifts on Directed Trees and Composition Operators in L 2-Spaces with Nondensely Defined Powers