Kre \u ın空间中算子的完备化与扩张

Lifshitz-Krein trace formula for Hirsch functiuonal calculus on Banach spaces

We give a simple definition of a spectral shift function for pairs of nonpositive operators on Banach spaces and prove trace formulas of Lifshitz-Kre\uın type for a perturbation of an operator monotonic (negative complete Bernstein) function of negative and nonpositive operators on Banach spaces induced by nuclear perturbation of an operator argument. The Lipschitzness of such functions is also investigated. The results may be regarded as a contribution to a perturbation theory for Hirsch functional calculus.

An indefinite range inclusion theorem for triplets of bounded linear operators on a Hilbert space

We study triplets of Hilbert space operators satisfying a certain inequality. A range inclusion theorem with norm estimate for those triplets is given with the language of Kre\uın space geometry and de Branges-Rovnyak space theory.

微扰行列式，谱移函数，迹恒等式，以及所有这些

Perturbation determinants, the spectral shift function, trace identities, and all that

讨论了M. G. Kre \u n理论的谱移函数的多维薛定谔算子以及该函数的具体性质，例如，它的高能量渐近。推导了薛定谔算子的迹恒等式。

与固定非半有界自伴算子相关的各种形式闭包

Various form closures associated with a fixed non-semibounded self-adjoint operator

本文研究了非半有界自伴算子形式闭包的非唯一性，建立了闭对称形式与Krein空间算子的一一对应，并指出闭对称形式可能比自伴算子携带更多信息。

扩张理论与非光滑边值问题的Krein型预解式

Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems

对于有界域$Ω\subset \mathbb{R}^n$上的强椭圆二阶算子$A$，多年来人们已经知道，当域和系数光滑时，如何将$A$的一般闭$L_2(Ω)$ -实现解释为表示边界条件（通常是非局部的）。本文的目的是将这种表示扩展到非光滑域和系数，包括Hölder $C^{\frac32+\varepsilon}$ -光滑的情况下，在这样一种方式，伪微分方法仍然可用于预解式的建设和椭圆度的考虑。我们展示了如何可以这样做的域与$B^\frac32_{p,2}$ -光滑和运营商的$H^1_q$ -系数，适当的$p>2(n-1)$和$q>n$。特别地，在这种非光滑情形下建立了Kre \u ın型预解式.允许某些无界域。

自伴算子的奇异扰动的Krein公式及其应用

A Krein-Like Formula for Singular Perturbations of Self-Adjoint Operators and Applications

给定一个自伴算子$A:D(A)\subseteq\calH\to\calH$和一个连续线性算子$τ:D(A)\to\X$，其值域为$ τ'\cap\calH' ={0}$，$\X$为Banach空间，我们显式构造了一族自伴算子$A^τ_Θ$，使得任何$A^τ_Θ$都与原$A$在$τ$的核上重合.这样一个族是通过给出一个Kre \u <$n-like公式得到的，其中亏空间的作用由对偶对$(\X,\X')$扮演;参数$Θ$属于从$\X'$到$\X$的对称算子空间。当$\X=\C$恢复“$\calH_{-2}$ -构造”的Kiselev和西蒙，因此，在某种程度上，我们的结果可以被视为它的无限秩的情况下的一个扩展。考虑到$\calH=L^2(\RE^n)$和$τ$是沿某个空子集的迹（限制）算子的情形，我们给出了非必要椭圆型伪微分算子奇异扰动的各种应用，从而统一和推广了已有的结果.

算子在Kreĭn空间上的自共轭算子的赫米特指数与分解

Hermitian indices and factorization of selfadjoint operators on a Kreĭn space

本文研究了自共轭算子在Kreĭn空间上的赫米特指数与分解，证明了在不依赖可分性的情况下，两种指数概念的一致性。

Factorized sectorial relations, their maximal sectorial extensions, and form sums

In this paper sectorial operators, or more generally, sectorial relations and their maximal sectorial extensions in a Hilbert space ${\mathfrak H}$ are considered. The particular interest is in sectorial relations $S$, which can be expressed in the factorized form \[ S=T^*(I+iB)T \quad \text{or} \quad S=T(I+iB)T^*, \] where $B$ is a bounded selfadjoint operator in a Hilbert space ${\mathfrak K}$ and $T:{\mathfrak H}\to{\mathfrak K}$ or $T:{\mathfrak K}\to{\mathfrak H}$, respectively, is a linear operator or a linear relation which is not assumed to be closed. Using the specific factorized form…

Selfadjoint extensions of relations whose domain and range are orthogonal

The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space ${\mathfrak H}_1$ to a Hilbert space ${\mathfrak H}_2$ are considered in the Hilbert space ${\mathfrak H}_1\oplus{\mathfrak H}_2$ that contains the graph of $R$. They will be described by $2 \times 2$ blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation $S$ in ${\mathfrak H}_1 \oplus {\mathfrak H}_2$ that is induced by $R$. Such a relation is characterized by the orthogonality property ${\rm dom\,} S \perp {\rm ran\,} S$ and it is nonnegative. All nonnegative self…

加性扰动下的自伴扩张

Self-Adjoint Extensions by Additive Perturbations

令$A_\N$是由$A$到$\N$的限制给出的对称算子，其中$A$是Hilbert空间$\H$上的自伴算子，$\N$是关于$A$的算子域$D(A)$上的图范数闭的线性稠密集。我们证明了$A_\N$的任何自伴扩张$A_Θ$使得$D(A_Θ)\cap D(A)=\N$可以被和$A_Θ=\A+T_Θ$相加分解，其中算子$\A$和$T_Θ$都取$D(A)$的强对偶中的值。算子$\A$是$A$对整个$\H$的封闭扩展，而$T_Θ$是根据$\N$和扩展参数$Θ$（一个与$A_\N$的亏空间同构（作为一个集合）的辅助希尔伯特空间上的自伴算子）的（抽象）边界条件明确地写的。给出了与Kre \u ın的预解式和冯·诺依曼的自伴扩张理论的明确联系。

On completion of the cone of CP linear maps with respect to the energy-constrained diamond norm

For a given positive operator $G$ we consider the cones of linear maps between Banach spaces of trace class operators characterized by the Stinespring-like representation with $\sqrt{G}$-bounded and $\sqrt{G}$-infinitesimal operators correspondingly. We prove the completeness of both cones w.r.t. the energy-constrained diamond norm induced by $G$ (as an energy observable) and the coincidence of the second cone with the completion of the cone of CP linear maps w.r.t. this norm. We show that the sets of quantum channels and quantum operations are complete w.r.t. the energy-constrained diamond no…