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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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