Preservation of the Unconditional Basis Property under Non-Self-Adjoint Perturbations of Self-Adjoint Operators

Abstract

Let T be a self-adjoint operator on a Hilbert space H with domain $\mathscr{D}(T)$. Assume that the spectrum of T is contained in the union of disjoint intervals Δ_k = [α_2k−1,α_2k], k ∈ ℤ, the lengths of the gaps between which satisfy the inequalities

$${\alpha _{2k + 1}} - {\alpha _{2k}}\geqslant b{\rm{|}}{\alpha _{2k + 1}} + {\alpha _{2k}}{{\rm{|}}^p}\;\;\;\;{\rm{for}}\;{\rm{some}}\;\;{\rm{b}} > 0,\;p \in [0,1).$$

Suppose that a linear operator B is p-subordinate to T, i.e.,

$$\mathscr{D}(B) \supset \mathscr{D}(T)\;\;\;{\rm{and}}\;\;\;\left\| {Bx} \right\|\leqslant b'{\left\| {Tx} \right\|^p}{\left\| x \right\|^{1 - p}} + M\left\| x \right\|\;\;\;\;{\rm{for}}\;{\rm{all}}\, x \in \mathscr{D}(T)$$

with some b′ > 0 and M ⩾ 0. Then, in the case of b > b′, for large |k| ⩾ N, the vertical lines γ_k = {∈ ℂ | Re λ = (α_2k + α_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 γ_k and γ_k+1 for |k| ⩾ N, and let Q be the Riesz projection onto the bounded remainder of the spectrum of A. The main result is as follows: The system {Q_k(H)}_|k|⩾Nof invariant subspaces together with the invariant subspace Q(H) forms an unconditional basis of subspaces in the space H. We prove also a generalization of this theorem to the case where any gap (α_2k,α_2k+1), k ∈ ℤ, may contain a finite number of eigenvalues of T.