Functional models for commutative systems of linear operators and de Branges spaces on a Riemann surface

Functional models are constructed for commutative systems $\{A_1,A_2\}$ of bounded linear non-self-adjoint operators which do not contain dissipative operators (which means that $\xi_1A_1+\xi_2A_2$ is not a dissipative operator for any $\xi_1$, $\xi_2\in\mathbb{R}$). A significant role is played here by the de Branges transform and the function classes occurring in this context. Classes of commutative systems of operators $\{A_1,A_2\}$ for which such a construction is possible are distinguished. Realizations of functional models in special spaces of meromorphic functions on Riemann surfaces are found, which lead to reasonable analogues of de Branges spaces on these Riemann surfaces. It turns out that the functions $E(p)$ and $\widetilde E(p)$ determining the order of growth in de Branges spaces on Riemann surfaces coincide with the well-known Baker-Akhiezer functions.
Bibliography: 11 titles.