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 contained in the union $\bigcup_{j\in J}\Delta_j$, $J\subseteq\mathbb{Z}$, of the 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 $\bigcup_{j\in J}U_b(\Delta_j)$ 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 $U_b(\Delta_j)$, $j\in J$. 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 $\mathcal{H}$.