Subnormal weighted shifts on directed trees whose nth powers have trivial domain

TL;DR

This paper constructs specific subnormal weighted shifts on directed trees and composition operators in L2-spaces demonstrating unusual domain properties of their powers, revealing new insights into operator behavior.

Contribution

It introduces novel examples of subnormal operators with powers exhibiting trivial or dense domains, advancing understanding of operator powers on directed trees and L2-spaces.

Findings

Existence of subnormal weighted shifts with trivial (n+1)th power domain

Construction of composition operators with similar properties

Demonstration of operator powers with contrasting domain behaviors

Abstract

It is shown that for any positive integer n there exists a subnormal weighted shift on a directed tree whose nth power is closed and densely defined while its (n + 1)th power has trivial domain. Similar result for composition operators in L2-spaces is established.