On functionsof class $\Pi$

D. Z. Arov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIII, Tome 135 (1984), pp. 5-30

Résumé

The class $\Pi$ of operator-valued functions $f$ satisfying the following properties is considered: 1) $f$ is meromorphic in $\mathbb C\setminus\mathbb T$; 2) the strong limits $\lim_{r\uparrow1}f(r\zeta)$ and $\lim_{r\downarrow1}f(r\zeta)$ 15) exist and coincide a. e. on $\mathbb T$; 3) $f(z)=f_2^{-1}(z)f_1(z)$, where $f_1$ is an operator-valued holomorphic function and $f_2$ is a complex-valued holomorphic function in $\mathbb C\setminus\mathbb T$. It is proved that every function in $\Pi$ is a compression of a $J$-inner function. Given $f\in\Pi$ all $J$-inner extensions of $f$ are described. The results obtained are applied to the realization problem of functions in $\Pi$ as transfer functions of linear systems.