A trace inequality for commuting \(d\)-tuples of operators (Q2140258)

The paper under review is devoted to variants of Berger-Shaw-type theorems for commuting tuples of hyponormal operators. The authors associate a determinant operator to the commutator matrix of a commuting tuple and discuss its role in the multivariate Berger-Shaw phenomenon. The determinant operator \(\det \big([T^*, T]\big)\) of a commuting pair \(T=(T_1, T_2)\) is given by \begin{align*} \mathrm{det} ([T^*, T]) = [T^*_1, T_1][T^*_2, T_2]+[T^*_2, T_2][T^*_1, T_1] \\ - [T^*_1, T_2][T^*_2, T_1]-[T^*_2, T_1][T^*_1, T_2]. \end{align*} It turns out that the above notion coincides with the generalized commutator introduced in [\textit{J. W. Helton} and \textit{R. E. Howe}, Acta Math. 135, 271--305 (1975; Zbl 0332.47010)] (see Theorem 3.4). One of the main results provides trace estimates for a family of commuting hyponormal operators (see, for example, Theorem 4.7).