This is an announcement for the paper “Unconditional bases of subspaces related to non-self-adjoint perturbations of self-adjoint operators” by A.K.Motovilovhttps://arxiv.org/find/math/1/au:+Motovilov_A/0/1/0/all/0/1, A.A.Shkalikovhttps://arxiv.org/find/math/1/au:+Shkalikov_A/0/1/0/all/0/1.
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 $\cup_{j\inJ}\Delta_j, J\subset\mathbb{Z}$, of segments $\Delta_j=[\alpha_j,\beta_j]\subset\mathbb{R}$ such that $\alpha_{j+1}>\beta_j$ and $$\inf_j(\alpha_{j+1}-beta_j)=d>0$$. 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 $\cup_{j\inJ} U_b(\Delta_j)$ 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 $U_b(\Delta_j)$. Our main result is as follows: The subspaces $\mathcalL}_j=Q_j(\mathcal{H}), j\in J$, form an unconditional basis in the whole space $\mathcal{H}$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.06296