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

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 $${\alpha }_{2k+1}-{\alpha }_{2k}\ge b|{\alpha }_{2k+1}+{\alpha }_{2k}{|}^p\text{ for some }b>0,p\in [0,1).$$ Suppose that a linear operator $B$ in $H$ is $p$-subordinated to $T$, i.e. $\mathcal{D}(B)\supset \mathcal{D}(T)$ and $\Vert Bx\Vert \le b^{\prime }\Vert Tx{\Vert }^p\Vert x{\Vert }^{1-p}+M\Vert x\Vert \text{ for all }x\in \mathcal{D}(T)$, with some $b^{\prime }>0$ and $M\ge 0$. Then the spectrum of the perturbed operator $A=T+B$ lies in the union of a rectangle in $\mathbb{C}$ and double parabola $P_{p,h}=\{\lambda \in \mathbb{C}||\mathrm{Im}\lambda |\le h|\mathrm{Re}\lambda {|}^p\}$, provided that $h>b^{\prime }$. The vertical strips ${\Omega }_k=\{\lambda \in \mathbb{C}||r_k-\mathrm{Re}\lambda |\le \delta r_k^p\}$, $r_k=({\alpha }_{2k}+{\alpha }_{2k+1})/2$, belong to the resolvent set of $T$, provided that $\delta <b-b^{\prime }$ and $|k|\ge N$ for $N$ large enough. For $|k|\ge N+1$, denote by ${\Pi }_k$ the curvilinear trapezoid formed by the lines $\mathrm{Re}\lambda =r_{k-1}$, $\mathrm{Re}\lambda =r_k$, 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 ${\bigcup }_{|k|\ge N+1}{\Pi }_k$. 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 $Q_k(H)$, $|k|\ge N+1$, together with the invariant subspace $Q_0(H)$ forms an unconditional basis of subspaces in the space $H$.

Main body of the paper is in Russian

References in corpus (1)

• Unconditional bases of subspaces related to non-self-adjoint perturbations of self-adjoint operators

Cited by in corpus (1)

• Local form subordination without a power decay and a criterion of Riesz basesness