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

Abstract

Assume that $T$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$ and that the spectrum of $T$ is confined in the union ${\bigcup }_{j\in J}{\Delta }_j$, $J\subseteq \mathbb{Z}$, of segments ${\Delta }_j=[{\alpha }_j,{\beta }_j]\subset \mathbb{R}$ such that ${\alpha }_{j+1}>{\beta }_j$ and $$\underset{j}{\inf }\left({\alpha }_{j+1}-{\beta }_j\right)=d>0.$$ If $B$ is a bounded (in general non-self-adjoint) perturbation of $T$ with $\Vert B\Vert =:b<d/2$ then the spectrum of the perturbed operator $A=T+B$ lies in the union ${\bigcup }_{j\in J}{U}_b({\Delta }_j)$ of the mutually disjoint closed $b$-neighborhoods $U_b({\Delta }_j)$ of the segments ${\Delta }_j$ in $\mathbb{C}$. Let $Q_j$ be the Riesz projection onto the invariant subspace of $A$ corresponding to the part of the spectrum of $A$ lying in $U_b\left({\Delta }_j\right)$, $j\in J$. Our main result is as follows: The subspaces ${\mathcal{L}}_j=Q_j(\mathcal{H})$, $j\in J$, form an unconditional basis in the whole space $\mathcal{H}$.

Cited by in corpus (1)

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