Low-Dimensional Thresholds and Optimal Weighted-Shift Bounds for the Aluthge Transform

Let \(A=U|A|\) be a complex matrix and let \(\Delta_\lambda(A)=|A|^\lambda U|A|^{1-\lambda}\), \(0<\lambda<1\), be its \(\lambda\)-Aluthge transform. Huang and Tam conjectured that the Frobenius norm of the self-commutator is contractive under \(\Delta_\lambda\). Following Zhang's recent counterexample, we study both the smallest dimension in which contractivity fails and the optimal constants for weighted cyclic shifts. We prove that the optimal dimension-two constant is \(C_{\lambda,2}=1\) for every \(0<\lambda<1\), whereas \(C_{\lambda,3}>1\) for every \(\lambda\ne\frac12\). Thus the minimal counterexample dimension is exactly three away from the symmetric parameter. For the usual Aluthge transform, we introduce cyclic constants \(\Gamma_n\) and path constants \(B_m\), prove an exact block decomposition principle, and determine\[ \Gamma_3=1,\qquad \Gamma_4=B_3=\sqrt{\frac{1+\sqrt2}{2}},\qquad B_4=\sqrt{q_0},\qquad B_5=\sqrt{q_5},\]where \(q_0\) and \(q_5\) are explicitly characterised algebraic numbers. In particular,\[ C_{1/2}\ge B_5=\sqrt{q_5}=1.13062037\ldots,\]which improves the previously known lower bound. The proofs combine Schur triangularisation, exact weighted-shift identities, block decomposition, recursive elimination of boundary variables, and convexity arguments.