Preserving of the unconditional basis property under non-self-adjoint perturbations of self-adjoint operators

TL;DR

This paper investigates how the unconditional basis property of a self-adjoint operator's invariant subspaces is preserved under certain non-self-adjoint perturbations, with spectrum localization and basis properties analyzed.

Contribution

It establishes conditions under which the invariant subspaces form an unconditional basis after non-self-adjoint perturbations of a self-adjoint operator.

Findings

Spectrum of perturbed operator is confined within specific regions in the complex plane.

Invariant subspaces form an unconditional basis in the Hilbert space.

Spectrum localization depends on the perturbation parameters and spectral gaps.

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 $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$, 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 $H$.