On orthosimilar systems in a~space of analytical functions and the problem of describing the dual space

V. V. Napalkov (Jr.)

Ufa mathematical journal, Tome 3 (2011) no. 1, pp. 30-41

Résumé

We consider an orthosimilar system with the measure $\mu$ in the space of analytical functions $H$ on the domain $G\subset\mathbb C$. Let $K_H(\xi,t)$, $\xi,t\in G$, be a reproduction kernel in the space $H$. We claim that a system $\{K_H(\xi,t)\}_{t\in G}$ is the orthosimilar system with the measure $\mu$ in the space $H$ if and only if the space $H$ coincides with the space $B_2(G,\mu)$. A problem of describing the dual space in terms of the Hilbert transform is considered. This problem is reduced to the problem of existence of a special orthosimilar system in $B_2(G,\mu)$. We prove that the space $\widetilde B_2(G,\mu)$ is the only space with a reproduction kernel and it consists of functions given on the domain $\mathbb C\setminus\overline G$ with an orthosimilar system $\{\frac1{(z-\xi)^2}\}_{\xi\in G}$ with the measure $\mu$.

Détail

Keywords: Bergman space, Hilbert spaces, reproducing kernel, orthosimilar system
Mots-clés : Hilbert transform.