The L. de Branges spaces and functional models of non-dissipative operators

The functional model for any bounded non-dissipative operator $A$ in Hilbert space $H$ with $\operatorname{rank}\Bigl(\dfrac{A-A^*}i\Bigr)=2$ has been constructed. This model is realized by the operator of multiplication on independent variable in the L. de Branges space of holomorphic functions. In difference with the L. de Branges space of entire functions the spaces of holomorphic in $\mathbb C$ functions with predefined singularities on the real axis have been studied. This allowed to construct the functional models for non-dissipative operators with real spectrum when $\operatorname{rank}\Bigl(\dfrac{A-A^*}i\Bigr)=2$.