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 ${\mathrm{\Delta }}_k=[{\alpha }_{2k-1},{\alpha }_{2k}]$, $k\in \mathbb{Z}$, and $${\alpha }_{2k+1}-{\alpha }_{2k}\ge b\mathrm{\mid }{\alpha }_{2k+1}+{\alpha }_{2k}{\mathrm{\mid }}^p\text{ for some }b>0,p\in [0,1)\mathrm{.}$$ Suppose that a linear operator $B$ in $H$ is $p$-subordinated to $T$, i.e. $\mathcal{D}(B)\supset \mathcal{D}(T)$ and $|Bx| \leq b’,|Tx|^p|x|^{1-p} +M|x| $\text{ for all }$ x\in \mathcal D(T)$,withsome$b’>0$and$M\geq 0$. Then the spectrum of the perturbed operator $A=T+B$ lies in the union of a rectangle in $\mathbb{C}$ and double parabola $P_{p,h} = \bigl{\lambda \in $\mathbb{C}$,\bigl|,|\mathop{\rm Im} \lambda|\leq h|\mathop{\rm Re} \lambda|^p\bigr}$,providedthat$h>b’$. The vertical strips $\Omega_k ={\lambda\in$\mathbb{C}$|,|r_k-{\rm Re},\lambda|\leq \delta r_k^p}$,$r_k =(\alpha_{2k}+\alpha_{2k+1})/2$, belong to the resolvent set of $T$, provided that $\delta <b-b^{\mathrm{\prime }}$ and $\mathrm{\mid }k\mathrm{\mid }\ge N$ for $N$ large enough. For $\mathrm{\mid }k\mathrm{\mid }\ge N+1$, denote by ${\mathrm{\Pi }}_k$ the curvilinear trapezoid formed by the lines $\mathrm{R}\mathrm{e}\lambda =r_{k-1}$, $\mathrm{R}\mathrm{e}\lambda =r_k$, and the boundary of the parabola $P_{p,h}$. Assume that $Q_0$ is the Riesz projection corresponding to the (bounded) part of the spectrum of $T$ that lies outside ${\bigcup }_{\mathrm{\mid }k\mathrm{\mid }\ge N+1}{\mathrm{\Pi }}_k$. And let $Q_k$, $\mathrm{\mid }k\mathrm{\mid }\ge N+1$, be the Riesz projection for the part of the spectrum of $T$ confined within ${\mathrm{\Pi }}_k$. Main result of the work consists in proving that the system of the invariant subspaces $Q_k(H)$, $\mathrm{\mid }k\mathrm{\mid }\ge N+1$, together with the invariant subspace $Q_0(H)$ forms an unconditional basis of subspaces in the space $H$.