On a conjecture of $\hat{I}»$-Aluthge transforms and Hilbert--Schmidt self-commutators

Abstract

Let $A$ be a complex square matrix, and write its polar decomposition as $A=U|A|$. For $0<\hat{I}»<1$, the $\hat{I}»$-Aluthge transform of $A$ is defined by $$\hat{I}{”}_{\hat{I}}»(A)=|A{|}^{\hat{I}}»U|A{|}^{1-\hat{I}»}.$$ In 2007, Huang and Tam conjectured that the Frobenius norm of the self-commutator is contractive under $\hat{I}{”}_{\hat{I}}»$: for every $0<\hat{I}»<1$, $$\Vert A^{\ast }A-A{A}^{\ast }{\Vert }_F\ge \Vert \hat{I}{”}_{\hat{I}}»(A{)}^{\ast }\hat{I}{”}_{\hat{I}}»(A)-\hat{I}{”}_{\hat{I}}»(A)\hat{I}{”}_{\hat{I}}»(A{)}^{\ast }{\Vert }_F.$$ If this inequality held, then the iterated self-commutator norms $$\{\Vert \hat{I}{”}_{\hat{I}}{»}^m(A{)}^{\ast }\hat{I}{”}_{\hat{I}}{»}^m(A)-\hat{I}{”}_{\hat{I}}{»}^m(A)\hat{I}{”}_{\hat{I}}{»}^m(A{)}^{\ast }{\Vert }_F{\}}_{m\in \mathbb{N}}$$ would form a nonincreasing sequence and necessarily converge to $0$. In this paper we provide a counterexample, thereby disproving the conjecture. We also obtain the quantitative bounds $$\sqrt{\frac{3}{2}}\le \underset{\begin{array}{c}A\in {\mathbb{M}}_n(\mathbb{C}),A^{\ast }A\ne A{A}^{\ast }\\ 0<\hat{I}»<1\end{array}}{\sup }\frac{\Vert \hat{I}{”}_{\hat{I}}»(A{)}^{\ast }\hat{I}{”}_{\hat{I}}»(A)-\hat{I}{”}_{\hat{I}}»(A)\hat{I}{”}_{\hat{I}}»(A{)}^{\ast }{\Vert }_F}{\Vert A^{\ast }A-A{A}^{\ast }{\Vert }_F}\le 2.$$