Existence Theorems for Momentum Representations Generalized in the Sense of Dzyadyk

In this paper, in particular, we prove that, for any sequence of complex numbers $\{c_n\}_{n=0}^\infty$, there exists a closed linear operator $A$ acting in the Hilbert space and two vectors $x$ and $y$ lying in the domains of definition of all powers of the operator $A$ for which the relation $c_n=(A^n x, y)$ holds. But if the series $\sum_{n=0}^\infty c_n z^n$ has radius of convergence $R > 0$, then in the representation $c_n=(A^nx,y)$, the operator $A$ can be chosen to be bounded with a spectral radius equal to $1/R$.