Joint hyponormality of Toeplitz pairs (Q2709432)

For bounded linear operators \(A\) and \(B\) on a Hilbert space \({\mathcal H}\), let \([A,B]\) be a commutator of \(A\) and \(B\). Given an \(n\)-tuple \({\mathbf T}= (T_1,\dots, T_n)\) of operators on \({\mathcal H}\), let \([{\mathbf T}^*,{\mathbf T}]= ([T^*_j, T_i])_{ij}\) \((1\leq i, j\leq n)\). If \([{\mathbf T}^*,{\mathbf T}]\geq 0\), then \({\mathbf T}\) is called jointly hyponormal. If the set \(\{\sum^n_{j=1} \alpha_j T_j:\alpha_j\in \mathbb{C})\) consists entirely of hyponormal operators, then \({\mathbf T}\) is called weakly hyponormal. For a single operator \(T\) on \({\mathcal H}\) is said to be \(k\)-hyponormal if \((T, T^2,\dots, T^k)\) is jointly hyponormal.NEWLINENEWLINENEWLINEOne of the main results in this paper is a complete characterization of jointly hyponormal trigonometric Toeplitz pairs, and the authors show this characterization can be extend to trigonometric Toeplitz \(n\)-tuples. This paper consists of the following five chapters:NEWLINENEWLINENEWLINEIn Chapter 1, they characterize the joint hyponormality of Toeplitz pairs with one coordinate a Toeplitz with an analytic polynomial symbol, and show that, for these pairs, joint hyponormality and weak hyponormality coincide.NEWLINENEWLINENEWLINEIn Chapter 2, they characterize the joint hyponormality of Toeplitz pairs in which both coordinates have trigonometric polynomial symbols, and show that the joint hyponormality of a trigonometric Toeplitz \(n\)-tuple can be detected by checking the joint hyponormality of all of its sub-pairs.NEWLINENEWLINENEWLINEIn Chapter 3, they consider the following question: Is every 2-hyponormal Toeplitz operator \(T_\varphi\) subnormal?, and show that every 2-hyponormal trigonometric Toeplitz operator is necessarily subnormal.NEWLINENEWLINENEWLINEIn Chapter 4, they introduce the notation of flatness for a Toeplitz pair and show that every jointly hyponormal trigonometric Toeplitz pair is necessarily flat, and discuss the Toeplitz extension problem of positive moment matrices.NEWLINENEWLINENEWLINEIn Chapter 5, they state some concluding remarks and open problems.