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 $\sum\nolimits_{n=0}^\infty {c_n z^n }$ has radius of convergence R > 0, then in the representation $c_n = (A^n x,y)$ , the operator A can be chosen to be bounded with a spectral radius equal to 1/R.