Hôpital Saint Charles

Uniqueness of the averaging measure in the three-matrix Sutter--Berta--Tomamichel trace inequality

Sutter, Berta, and Tomamichel asked whether the averaging density \(\beta_0\) in their three-matrix trace inequality could be replaced by another probability distribution independent of the matrices. We show that it cannot. In every fixed dimension \(d\geq2\), if a Borel probabil...

Uniqueness of the averaging measure in the three-matrix Sutter--Berta--Tomamichel trace inequality

Sutter, Berta, and Tomamichel asked whether the averaging density \(\beta_0\) in their three-matrix trace inequality could be replaced by another probability distribution independent of the matrices. We show that it cannot. In every fixed dimension \(d\geq2\), if a Borel probabil...

Uniqueness of the averaging measure in the three-matrix Sutter--Berta--Tomamichel trace inequality

Sutter, Berta, and Tomamichel asked whether the averaging density \(\beta_0\) in their three-matrix trace inequality could be replaced by another probability distribution independent of the matrices. We show that it cannot. In every fixed dimension \(d\geq2\), if a Borel probabil...

The one-parameter Böttcher--Wenzel conjecture holds for \(3\times 3\) nilpotent matrices

Chru\'sci\'nski, Kimura, Ohno, and Singal conjectured a sharp bound for a one-parameter deformation of the B\"ottcher\textendash Wenzel functional on pairs of complex matrices, with an explicit constant $c(q)$ depending on a real parameter $q$. This paper proves the conjecture fo...

The one-parameter Böttcher--Wenzel conjecture holds for \(3\times 3\) nilpotent matrices

Chru\'sci\'nski, Kimura, Ohno, and Singal conjectured a sharp bound for a one-parameter deformation of the B\"ottcher\textendash Wenzel functional on pairs of complex matrices, with an explicit constant $c(q)$ depending on a real parameter $q$. This paper proves the conjecture fo...

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'...

The complete singular-value spectrum of a twisted commutator with a rank-one factor and an application to the weighted Böttcher–Wenzel inequality

Let \(D\) be positive definite Hermitian and let \(T_X:Y\mapsto XDY-YDX\) be the associated twisted commutator on \(M_n(\mathbb C)\) with the Frobenius norm. For a rank-one factor \(X=\sigma uv^*\), an orthogonal block decomposition gives the exact operator norm \[\lVert T_X\rVer...

The complete singular-value spectrum of a twisted commutator with a rank-one factor and an application to the weighted Böttcher–Wenzel inequality

Let \(D\) be positive definite Hermitian and let \(T_X:Y\mapsto XDY-YDX\) be the associated twisted commutator on \(M_n(\mathbb C)\) with the Frobenius norm. For a rank-one factor \(X=\sigma uv^*\), an orthogonal block decomposition gives the exact operator norm \[\lVert T_X\rVer...