an:06060965
Zbl 1248.47029
Curto, Ra??l E.; Hwang, In Sung; Lee, Woo Young
Hyponormality and subnormality of block Toeplitz operators
EN
Adv. Math. 230, No. 4-6, 2094-2151 (2012).
00303526
2012
j
47B35 47B20 47A13 30H10 47A20 47A57
block Toeplitz operator; hyponormal operator; square-hyponormal operator; subnormal operator; bounded type function; rational function; trigonometric polynomial; Toeplitz subnormal completion problem
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.
Alexei Yu. Karlovich (Lisboa)