A. A. Tarasenko

In this work completely continious nondissipative operators with two-dimensional imaginary parts, acting in separable Hilbert space are studied. The criteria of completeness and unconditional basis property of root vectors of such operators are obtained. The results are formulated in terms of characteristic matrix-valued functions of nonselfadjoint operators and proved using analysis of functional models in de Branges spaces.

Let a sequence $\left\{v_k \right\}^{+\infty}_{-\infty}\in l_2$ and a real sequence $\left\{\lambda_k \right\}^{+\infty}_{-\infty}$ such that $\left\{\lambda_k^{-1} \right\}^{+\infty}_{-\infty}\in l_2$, and an orthonormal basis $\left\{e_k \right\}^{+\infty}_{-\infty}$ of a Hilbert space be given. We describe a sequence $M=\left\{\mu_k \right\}^{+\infty}_{-\infty}$, $M\cap \mathbb{R}=\varnothing$, such that the families $$ f_k = \sum\limits_{j\in\mathbb{Z}} {v_j\left(\lambda_j-\bar{\mu}_k \right)^{-1}}e_k, \quad k\in \mathbb{Z} $$ form an unconditional basis in $\mathfrak{H}$.