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Hyponormality and subnormality of block Toeplitz operators. (English) Zbl 1248.47029
An operator \(T\) on a Hilbert space is said to be normal if \(T^*T-TT^*=0\), hyponormal if \(T^*T-TT^*\geq 0\), and subnormal if it has a normal extension. The paper is devoted to hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space \(H^2\) of the unit circle. A function \(\varphi\in L^\infty\) is said to be of bounded type (or in the Nevanlinna class) if there are analytic functions \(\psi_1,\psi_2\in H^\infty\) such that \(\varphi=\psi_1/\psi_2\) almost everywhere on the unit circle. The first main result of the paper is a criterion for the hyponormality of block Toeplitz operators with bounded type symbols. The second main result is related to the Halmos problem: is every subnormal Toeplitz operator either normal or analytic? It is shown that, if \(\Phi\) is a matrix-valued rational function whose co-analytic part has a coprime factorization, then every hyponormal Toeplitz operator \(T_\Phi\) whose square is also hyponromal must be either normal or analytic. Third, using the subnormality theory of block Toeplitz operators, the authors give an answer to a Toeplitz subnormal completion problem. Finally, some open problems are listed.

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B20 Subnormal operators, hyponormal operators, etc.
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
30H10 Hardy spaces
47A20 Dilations, extensions, compressions of linear operators
47A57 Linear operator methods in interpolation, moment and extension problems
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