Aluthge transforms of weighted shifts on directed trees

TL;DR

This paper extends the Aluthge transform to unbounded operators, provides a formula for weighted shifts on directed trees, and constructs examples illustrating unique properties of the transform.

Contribution

It introduces a generalized Aluthge transform for unbounded operators and constructs novel examples demonstrating its unusual properties.

Findings

Aluthge transform of unbounded operators can have trivial domain.

Constructed hyponormal operator with non-closable Aluthge transform.

Extended Aluthge transform to weighted shifts on directed trees.

Abstract

Aluthge transform of a bounded operator is generalized to the case of unbounded one. A formula for the Aluthge transform of a weighted shift on a directed tree is established and it is used to construct an example of a hyponormal operator whose Aluthge transform has trivial domain. It is proven that such an example can be also constructed in the class of composition operators. It is also shown that Aluthge transform of a closed, densely defined operator is not necessarily closable.