On a class of unconditional bases in Hilbert spaces and on the~problem of similarity of dissipative Volterra operators

G. M. Gubreev

Sbornik. Mathematics, Tome 77 (1994) no. 1, pp. 93-126

Résumé

Let $B$ be a completely nonselfadjoint dissipative Volterra operator acting in a separable Hilbert space $\mathfrak Y$ whose resolvent $(I-\lambda B)^{-1}$ has finite exponential type. Further, let $\mathfrak{L}=(B-B^*)\mathfrak Y$, $y\in\mathfrak{L}$, and $y(\lambda)=(I-\lambda B)^{-1}y$. In this article conditions are determined on the operator $B$, the vector $y$, and the sequence $\Lambda=\{\lambda_k\}_{-\infty}^{+\infty}$ under which the family $$ \{y(\lambda_k):\lambda_k\in \Lambda\}, \qquad \inf_{\lambda_k}\operatorname{Im}\lambda_k>0, $$ forms an unconditional basis in the space $\mathfrak Y$. Moreover, a new approach is considered for the problem of similarity of dissipative Volterra operators, based on a study of the basis properties of this system of vectors.