Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Preserving of the unconditional basis property under non-self-adjoint
  perturbations of self-adjoint operators   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 $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$,
and $$ \alpha_{2k+1}-\alpha_{2k} \geq b|\alpha_{2k+1}+\alpha_{2k}|^p\quad
\text{ for some }\,b>0,\,p\in[0,1). $$ Suppose that a linear operator $B$ in
$H$ is $p$-subordinated to $T$, i.e. $\mathcal D(B) \supset\mathcal D(T)$ and
$\|Bx\| \leq b'\,\|Tx\|^p\|x\|^{1-p} +M\|x\| \text{\, for all } x\in \mathcal
D(T)$, with some $b'>0$ and $M\geq 0$. Then the spectrum of the perturbed
operator $A=T+B$ lies in the union of a rectangle in $\mathbb{C}$ and double
parabola $P_{p,h} = \bigl\{\lambda \in \mathbb{C}\,\bigl|\,|\mathop{\rm Im}
\lambda|\leq h|\mathop{\rm Re} \lambda|^p\bigr\}$, provided that $h>b'$. The
vertical strips $\Omega_k =\{\lambda\in\mathbb{C}|\,|r_k-{\rm Re}\,\lambda|\leq
\delta r_k^p\}$, $r_k =(\alpha_{2k}+\alpha_{2k+1})/2$, belong to the resolvent
set of $T$, provided that $\delta <b -b'$ and $ |k|\geq N$ for $N$ large
enough. For $|k|\ge N+1$, denote by $\Pi_k$ the curvilinear trapezoid formed by
the lines ${\rm Re}\,\lambda = r_{k-1}$, ${\rm Re}\,\lambda = r_{k}$, and the
boundary of the parabola $P_{p,h}$. Assume that $Q_0$ is the Riesz projection
corresponding to the (bounded) part of the spectrum of $T$ that lies outside
$\bigcup_{|k|\geq N+1}\Pi_k$. And let $Q_k$, $|k|\ge N+1$, be the Riesz
projection for the part of the spectrum of $T$ confined within $\Pi_k$. Main
result of the work consists in proving that the system of the invariant
subspaces $Q_k(H)$, $|k|\geq N+1$, together with the invariant subspace
$Q_0(H)$ forms an unconditional basis of subspaces in the space $H$.

37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn