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

TL;DR

This paper proves that under certain spectral gap conditions, the invariant subspaces of a perturbed self-adjoint operator form an unconditional basis, extending classical spectral theory to non-self-adjoint perturbations.

Contribution

It establishes that the invariant subspaces associated with spectral neighborhoods form an unconditional basis for the Hilbert space under bounded non-self-adjoint perturbations.

Findings

Invariant subspaces form an unconditional basis.

Spectral perturbation remains within controlled neighborhoods.

Main result applies to operators with spectral gaps.

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 $$ \inf_{j} \left(\alpha_{j+1}-\beta_j\right) = d > 0. $$ If $B$ is a bounded (in general non-self-adjoint) perturbation of $T$ with $\|B\|=:b<d/2$ 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 $\mathcal H$.