Mind Duggal Transforms

TL;DR

This paper demonstrates that unlike the Aluthge transform, the Duggal transform does not necessarily preserve the complex symmetric property of operators, correcting a previous misconception.

Contribution

It provides a counterexample showing the Duggal transform does not always inherit complex symmetry, clarifying a previously claimed property.

Findings

Duggal transform isn't always complex symmetric when the original operator is.

The paper corrects a misconception in prior literature.

Provides explicit examples illustrating the failure of property inheritance.

Abstract

It is known that if an operator $T$ is complex symmetric then its Aluthge transform is also complex symmetric. This Note is devoted to showing that the Duggal transform doesn't inherit this property. For instance, we'll show that the Duggal transform isn't always complex symmetric when $T$ is, as it was claimed in \cite{Ga}.