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Preserving of the unconditional basis property under non-self-adjoint perturbations of self-adjoint operators
A. K. Motovilova,
A. A. Shkalikovb a Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics
b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
Let
$T$ be a self-adjoint operator in a Hilbert space
$H$ with domain
$\mathcal D(T)$. Assume that the spectrum of
$T$ is confined in the union of disjoint intervals
$\Delta_k =[\alpha_{2k-1},\, \alpha_{2k}]$,
$k\in \mathbb{Z}$, the lengths of the gaps between which satisfy inequalities
\begin{equation*}
\alpha_{2k+1}-\alpha_{2k} \geqslant b
|\alpha_{2k+1}+\alpha_{2k}|^p\quad \text{ for some }\, b\ge 0,\,
p\in[0,1).
\end{equation*}
Suppose that a linear operator
$B$ is
$p$-subordinated to
$T$, i.e.
$\mathcal D(B) \supset\mathcal D(T)$ and $\|Bx\| \leqslant b\,\|Tx\|^p\|x\|^{1-p} +M\|x\| \text{\, for all } x\in \mathcal D(T)$, with some
$b\ge0$ and
$M\geqslant 0$. Then in the case of
$b\ge b$, for large
$|k|\geqslant N$, the vertical lines $\gamma_k = \{\lambda\in\mathbb{C}\,| \mathop{\rm Re} \lambda = (\alpha_{2k} + \alpha_{2k+1})/2\}$ lie in the resolvent set of the perturbed operator
$A=T+B$. Let
$Q_k$ be the Riesz projections associated with the parts of the spectrum of
$A$ lying between the lines
$\gamma_k$ and
$\gamma_{k+1}$ for
$|k|\geqslant N$, and let
$Q$ be the Riesz projection for the remainder of the spectrum of
$A$. Main result is as follows: The system of the invariant subspaces
$\{Q_k(H)\}_{|k|\geqslant N}$ together with the invariant subspace
$Q(H)$ forms an unconditional basis of subspaces in the space
$H$. We also prove a generalization of this theorem to the case where any gap
$(\alpha_{2k},\,\alpha_{2k+1})$,
$k\in\mathbb{Z}$, may contain a finite number of eigenvalues of
$T$.
Keywords:
Riesz basis, unconditional basis of subspaces, non-self-adjoint perturbations.
UDC:
517.984
MSC: 47A55,
47A15 Received: 18.11.2018
Revised: 13.05.2019
Accepted: 16.05.2019
DOI:
10.4213/faa3632
Fulltext:
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