On a conjecture of $\lambda$-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 $0<\lambda<1$, the $\lambda$-Aluthge transform of $A$ is defined by $$ \Delta_\lambda(A)=|A|^\lambda U|A|^{1-\lambda}. $$ In 2007, Huang and Tam conjectured that the Frobenius norm of the self-commutator is contractive under $\Delta_\lambda$: for every $0<\lambda<1$, $$ \|A^*A-AA^*\|_{F} \ \ge\ \|\Delta_\lambda(A)^*\Delta_\lambda(A)-\Delta_\lambda(A)\Delta_\lambda(A)^*\|_{F}. $$ If this inequality held, then the iterated self-commutator norms $$ \Bigl\{\bigl\|\Delta_\lambda^{\,m}(A)^*\Delta_\lambda^{\,m}(A) -\Delta_\lambda^{\,m}(A)\Delta_\lambda^{\,m}(A)^*\bigr\|_F\Bigr\}_{m\in\mathbb N} $$ 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 $$ \sqrt{\frac32}\ \le\ \sup_{\substack{A\in\mathbb{M}_n(\mathbb{C}),\ A^*A\neq AA^*\\ 0<\lambda<1}} \frac{\|\Delta_\lambda(A)^*\Delta_\lambda(A)-\Delta_\lambda(A)\Delta_\lambda(A)^*\|_F}{\|A^*A-AA^*\|_F} \ \le\ 2. $$