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

Let $T$ be a self-adjoint operator in a Hilbert space $H$ with domain $\mathcal D(T)$. Assume that the spectrum of $T$ is confined in the union of disjoint intervals $\Delta_k =[\alpha_{2k-1},\, \alpha_{2k}]$, $k\in \mathbb{Z}$, the lengths of the gaps between which satisfy inequalities
\begin{equation*} \alpha_{2k+1}-\alpha_{2k} \geqslant b |\alpha_{2k+1}+\alpha_{2k}|^p\quad \text{ for some }\, b\ge 0,\, p\in[0,1). \end{equation*}
Suppose that a linear operator $B$ is $p$-subordinated to $T$, i.e. $\mathcal D(B) \supset\mathcal D(T)$ and $\|Bx\| \leqslant b\,\|Tx\|^p\|x\|^{1-p} +M\|x\| \text{\, for all } x\in \mathcal D(T)$, with some $b\ge0$ and $M\geqslant 0$. Then in the case of $b\ge b$, for large $|k|\geqslant N$, the vertical lines $\gamma_k = \{\lambda\in\mathbb{C}\,| \mathop{\rm Re} \lambda = (\alpha_{2k} + \alpha_{2k+1})/2\}$ lie in the resolvent set of the perturbed operator $A=T+B$. Let $Q_k$ be the Riesz projections associated with the parts of the spectrum of $A$ lying between the lines $\gamma_k$ and $\gamma_{k+1}$ for $|k|\geqslant N$, and let $Q$ be the Riesz projection for the remainder of the spectrum of $A$. Main result is as follows: The system of the invariant subspaces $\{Q_k(H)\}_{|k|\geqslant N}$ together with the invariant subspace $Q(H)$ forms an unconditional basis of subspaces in the space $H$. We also prove a generalization of this theorem to the case where any gap $(\alpha_{2k},\,\alpha_{2k+1})$, $k\in\mathbb{Z}$, may contain a finite number of eigenvalues of $T$.