We generalize rank one singular nonsymmetric perturbations of a self-adjoint operator from the class H �1 to the case of a finite rank. The definition and description of the resolvent are presented. The main assertions made in the paper are illustrated by a specific example.
УДК 517.9
Узагальнено зв’язок класичної проблеми моментiв iз спектральною теорiєю матриць Якобi. Наведено розв’язок двовимiрної напiвсильної проблеми моментiв та запропоновано аналог матриць типу Якобi, що вiдповiдає двовимiрнiй напiвсильнiй проблемi моментiв, та вiдповiдну систему полiномiв, ортогональних вiдносно мiри iз компактним носiєм на дiйснiй площинi.
The singular nonsymmetric rank one perturbation of
a self-adjoint operator from classes ${\mathcal H}_{-1}$ and ${\mathcal H}_{-2}$ was considered for the first time in works by
Dudkin M.E. and Vdovenko T.I. \cite{k8,k9}. In the mentioned papers, some properties of the point spectrum are described,
which occur during such perturbations.
This paper proposes generalizations of the results presented in \cite{k8,k9} and \cite{k2} in the case of
nonsymmetric class ${\mathcal H}_{-2}$ perturbations of finite rank.
That is, the formal expression of the following is considered
\begin{equation*}
\tilde A=A+\sum \limits_{j=1}^{n}\alpha_j\langle\cdot,\omega_j\rangle\delta_j,
\end{equation*}
where $A$ is an unperturbed self-adjoint operator on a separable Hilbert space
${\mathcal H}$, $\alpha_j\in{\mathbb C}$, $\omega_j$, $\delta_j$, $j=1,2, ..., n<\infty$ are
vectors from the negative space ${\mathcal H}_{-2}$ constructed by the operator $A$,
$\langle\cdot,\cdot\rangle$ is the dual scalar product between positive and negative spaces.
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