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

Let \(A=U|A|\) be a complex matrix and let \(Δ_λ(A)=|A|^λ U|A|1-λ\), \(0<λ<1\), be its \(λ\)-Aluthge transform. Huang and Tam conjectured that the Frobenius norm of the self-commutator is contractive under \(Δ_λ\). 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 \(Cλ,2=1\) for every \(0<λ<1\), whereas \(Cλ,3>1\) for every \(λ≠12\). Thus the minimal counterexample dimension is exactly three away from the symmetric parameter. For the usual Aluthge transform, we introduce cyclic constants \(Γ_n\) and path constants \(B_m\), prove an exact block decomposition principle, and determine\[ Γ_3=1, Γ_4=B_3=√1+√2/2, B_4=√q_0, B_5=√q_5,\]where \(q_0\) and \(q_5\) are explicitly characterised algebraic numbers. In particular,\[ C1/2≥ B_5=√q_5=1.13062037…,\]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.