On orthosimilar systems in a space of analytical functions and the problem of describing the dual space
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 ${\mathbb C}\backslash \overline G$ with an orthosimilar system $\{\tfrac{1}{(z-\xi)^2}\}_{\xi\in G}$ with the measure $\mu$.