On realization of the Krein-Langer class Nk of matrix-valued functions in Hilbert spaces with indefinite metric

Dissipative operators in Krein Space: invariant subspaces and properties of restrictions

We prove that a dissipative operator in Krein space possesses a maximal non-negative invariant subspace provided that this operator admits matrix representation with respect to canonical decomposition of the space and the right upper entry of the operator matrix is relatively compact with respect to the right lower entry. Under additional condition when the left upper and left lower operators are bounded (the so-called "Langer condition") this result was proved (in the increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. The Langer condition is replaced in the paper by a m…

The Classical Moment Problem and Generalized Indefinite Strings

We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. Namely, we introduce the class of Krein-Langer strings and show that there is a bijective correspondence between moment sequences and this class of generalized indefinite strings. This result can be viewed as a complement to the classical results of M. G. Krein on the connection between the Stieltjes moment problem and Krein-Stieltjes strings and I. S. Kac on the connection between the Hamburger moment problem and 2x2 canonical systems with Hamburger Hamiltonians.

Pontryagin 空间中对称线性关系的 L-预解式

L-resolvents of symmetric linear relations in Pontryagin spaces

本文在Pontryagin空间框架下，建立具有不合适规范的对称线性关系的边界三元组理论与Krein–Saakyan L-预解矩阵理论的联系，推广L-预解矩阵的边界算子表示公式，并应用于典范系统生成的极小线性关系。

广义不定弦的迹公式与逆谱理论

Trace formulas and inverse spectral theory for generalized indefinite strings

本文研究了广义不定弦的逆谱理论，证明了相关的 Szegő 型定理，完整刻画了特定系数类的谱测度集合，并将其应用于保守 Camassa-Holm 流，证明了低正则性阶梯型初始条件下全局弱解的存在性。

Bounds on the non-real spectrum of differential operators with indefinite weights

Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and $\infty$ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm-Liouville and second order elliptic partial differential operators with i…

Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x

We consider a singular Sturm-Liouville expression with the indefinite weight sgn x. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of $\infty$. Moreover, in this situation, the point $\infty$ is a regular critical point. We construct an operator $A=(\sgn x)(-d^2/dx^2+q)$ with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators.