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

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}$$,ofsegments$\Delta_j=[\alpha_j, \beta_j]\subset$\mathbb{R}$$suchthat$\alpha_{j+1}>\beta_j$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 $\mathrm{\parallel }B\mathrm{\parallel }=:b<d\mathrm{/}2$ then the spectrum of the perturbed operator $A=T+B$ lies in the union $\bigcup_{j\in J} U_{b}(\Delta_j)$ofthemutuallydisjointclosed$b$-neighborhoods$U_{b}(\Delta_j)$ofthesegments$\Delta_j$in$$\mathbb{C}$$\mathrm{.}Let$Q_j$ be the Riesz projection onto the invariant subspace of $A$ corresponding to the part of the spectrum of $A$ lying in $U_b\left({\mathrm{\Delta }}_j\right)$, $j\in J$. Our main result is as follows: The subspaces ${\mathcal{L}}_j=Q_j(\mathcal{H})$, $j\in J$,formanunconditionalbasisinthewholespace$\mathcal H$.