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