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.

中文翻译：

---

自伴算子的非自伴扰动下无条件基性的保存

设T为Hilbert空间H上域\（\ mathscr {D}（T）\）的自伴随算子。假设的频谱Ť被包含在不相交间隔Δ的联合_ķ = [ α _{2 ķ -1}，α _{2 ķ} ] ķ ∈ℤ，间隙的长度在它们之间满足不等式$$ {\阿尔法_ { 2k + 1}}-{\ alpha _ {2k}} \ geqslant b {\ rm {|}} {\ alpha _ {2k + 1}} + {\ alpha _ {2k}} {{\ rm {|} } ^ p} \; \; \; \; {\ rm {for}} \; {\ rm {some}} \; \; {\ rm {b}}> 0，\; p \ in [0， 1）。$$假设线性算子B是p-从属于T，即$$ \ mathscr {D}（B）\ supset \ mathscr {D}（T）\; \; \; {\ rm {and}} \; \; \; \ left \ | {Bx} \ right \ | \ leqslant b'{\ left \ | {Tx} \ right \ | ^ p} {\ left \ | x \ right \ | ^ {1-p}} + M \ left \ | x \ right \ | \; \; \; \; \; {\ rm {for}} \; {\ rm {all}} \，x \ in \ mathscr {D}（T）$$，且b' > 0和中号⩾0。然后，在的情况下，b> b' ，对于大| k | ⩾ Ñ，竖直线γ _ķ = {∈ℂ| 再λ =（α _{2 ķ} + α _{2 ķ 1}）/ 2}横亘在解集扰动算子的甲= Ť +乙。让Q _ķ与的光谱的各部分相关联的中Riesz突起甲行间位于γ _ķ和γ _{ķ 1}为| k | ⩾ Ñ，并让Q是中Riesz投影到的频谱的有界剩余甲。主要结果是，如下所示：该系统{ Q _ķ（ħ）} _{| K |⩾ Ñ}不变子空间与不变子空间Q一起的（ħ）形成的子空间中的空间H.无条件地我们还证明这个定理的推广到其中的任何间隙的情况下（α _{2 ķ}，α _{2 ķ 1}），ķ＆Element; ℤ，可以含有的本征值的有限数量的Ť。