Self-Adjoint Extension of Symmetric Maps
H. N. Friedel
从实希尔伯特空间到实希尔伯特空间的稠密定义对称线性映射扩张到自伴映射。扩展通过Riesz表示来表示。对于强单调映射的Friedrichs扩张，自伴扩张是唯一的，并且等于给定映射的闭包。
A densely-defined symmetric linear map from/to a real Hilbert space extends to a self-adjoint map. Extension is expressed via Riesz representation. For a case including Friedrichs extension of a strongly monotone map, self-adjoint extension is unique, and equals closure of the given map.
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Maps on self-adjoint operators preserving some relations related to commutativity
本文研究了 Hilbert 空间上自伴有界算子间双向保持涉及算子差与交换性或拟交换性三元关系的映射，证明此类映射由酉或反酉等价性及标量算子加性扰动实现。
Linear maps between C*-algebras whose adjoints preserve extreme points of the dual ball
本文给出了从一个$C^\ast$ % -代数到另一个$C^\ast$ % -代数的线性算子的一个结构特征，其伴随算子将对偶球的极值点映射到极值点上.我们表明，直到一个$\ast$ -同构，这样的地图允许分解成一个退化和一个非退化的部分，其中的非退化部分出现作为约旦$\ast$ -态射，其次是一个"旋转“，然后减少。在伴随保持纯态的映射的情况下，退化部分不出现，而“旋转”只是恒等式。在这种情况下，关于这种纯状态保持映射的结果依赖于Størmer [Stø 2; 5.6 \& 5.7]。最后，我们考虑了带有“极值点重叠”伴随的映射在某些特定的$C^\ast$ -代数上的作用。
On isometry groups of self-adjoint traceless and skew-symmetric matrices
本文研究自伴无迹矩阵空间上的等距。我们计算组的等距相对于任何酉相似不变范数。这完善和推广了Nagy关于Schatten $p$ -范数等距的结果.此外，我们指出，我们的证明技术可以应用到获得一个旧的结果，关于对称矩阵的等距。
$C$-self-adjoint contractive extensions of $C$-symmetric non-densely defined contractions
本文研究希尔伯特空间中非稠定$C$-对称压缩算子的$C$-自伴压缩延拓，得到全部延拓的参数化形式，并应用该结果证明了Glazman提出的稠定$C$-对称耗散算子存在极大耗散$C$-自伴延拓的结论。
On $C$-Symmetric and $C$-Self-adjoint Unbounded Operators on Hilbert Space
本文研究希尔伯特空间上关于共轭$C$的$C$-对称与$C$-自伴无界算子，给出了$C$-自伴扩张的完全描述、基于拟解析向量的判据及极分解刻画。
Functional Calculus for definitizable self-adjoint linear relations on Krein spaces
本文发展了Krein空间上自伴可定义线性关系的谓词演算$ϕ\mapsto ϕ(A)$.这个谓词演算是希尔伯特空间情形中的$ϕ\mapsto \int ϕ\, dE$的适当模拟。它还包括频谱定理的自伴可定义运营商的克莱因空间显示存在的频谱投影。
Self-adjoint boundary conditions for the prolate spheroid differential operator
考虑有限对称区间上的形式长椭球微分算子，描述了其自伴边界条件。这些边界条件中只有一个对应于一个自伴微分算子，该算子与在所考虑的有限对称区间上截断的傅立叶算子可交换。
Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with equal deficiency indices $n_\pm (A)\leq\infty$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an exit space extension of $A$; such an extension is called finite-codimensional if $\dim (\wt\gH\ominus\gH)< \infty$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of exit space extensions $\wt A=\wt A^*$. For a certain class of extensions $\wt A$ we parameterize the compressions $C (\wt A)$ by means of abstract boundary conditions. This enables us to characterize vari…
Symmetric pairs of unbounded operators in Hilbert space, and their applications in mathematical physics
在以前的论文中，作者介绍了对称算子对的概念，作为计算对称算子的自伴扩张的一种方法。简而言之，对称对由两个密集定义的线性算子$A$和$B$组成，其中有$A \subseteq B^\star$和$B \subseteq A^\star$。在本文中，我们将通过例子表明，对称对可以用来推导算子的可闭性，有时甚至计算伴随。特别地，证明了随机分析的Malliavin导数和Skorokhod积分是可闭的，且闭包是相互伴随的.我们还证明了Tomita-Takesaki理论的基本对合是可闭的，并且它们的闭包是相互伴随的。本文还讨论了Laplace算子和包含算子在无限图上的有限能量函数中的应用，其中Laplace算子和包含算子构成一个对称对。
On extensions of $J$-skew-symmetric and $J$-isometric operators
本文证明了Hilbert空间$H$中的每个稠定$J$ -反对称算子（或每个具有$\overline{D(A)}=\overline{R(A)}=H$的$J$ -等距算子）在Hilbert空间$\widetilde H\supseteq H$中有$J$ -反自伴（或$J$ -酉）扩张.我们遵循Galindo在~[A.~]中的思想。Galindo，关于$J$的存在性-$J$的自伴扩展-具有伴随的对称算子，《纯粹数学和应用数学通讯》，第十五卷，423-425（1962）]，并进行了必要的修改。
A criterion for essential self-adjointness of a symmetric operator defined by some infinite hermitian matrix with unbounded entries
我们将考虑一个双无限的、厄米的、复元素矩阵$A=[a_{x,y}]_{x,y\in\mathbb Z}$，其中有$a_{x,y}^*=a_{y,x}$，$x,y\in\mathbb Z$。假设矩阵几乎是有限带宽的，即存在一个整数$n> 0$和指数$γ\in[0,1)$，使得对所有$z>n\langle x\rangle^γ$的$ a_{x,x+z}=0$和一行的$\ell_1$范数的增长比对$|x|\gg1$的$|x|^{1-γ}$慢，即$\lim_{|x|\to+\infty}| x|^{γ-1}\sum_{y}|a_{xy}|=0$，我们证明了定义在紧支撑序列上的相应对称算子在$\ell_2(\mathbb Z)$中本质上是自伴的.在$γ=0$（所谓的$(nJ)$ -矩阵）的情况下，我们证明了存在$c_*>0$，仅依赖于$n$，使得条件$\limsup_{|x|\to+\infty}| x|^{-1}\sum_{y}|a_{xy}|\le c_*$足以得出本质自伴性的结论。
We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both real and complex spaces. As an application of the results and the construction we consider positive self-adjoint extensions of the modulus square operator $T^*T$ of a densely defined linear transformation $T$ and bounded self-adjoint extensions of a symmetric operator. Krein's results on the uniqueness of positive (respectively, norm preserving) self-adjoint e…
Maps on real Hilbert spaces preserving the area of parallelograms and a preserver problem on self-adjoint operators
本文首先用$2\leq d<\infty$描述了$\mathbb{R}^d$上的所有（不一定是线性或双射）变换，这些变换保持了任意两个向量所张成的平行四边形的面积。我们还刻画了任意实希尔伯特空间上保持后一个量的双射（不一定是线性的）。这回答了Rassias和Wagner提出的一个问题，它可以被认为是著名的实希尔伯特空间上的Wigner定理的一个变体，该定理在量子力学中起着重要作用。因此，我们解决了Molnár和Timmermann的一个问题，该问题仅在二维情况下顽固地保持开放。最后，这个二维的结果将被应用，以加强他们的定理在更高的维度。
Symmetric pairs and self-adjoint extensions of operators, with applications to energy networks
利用对称算子对，给出了Hilbert空间$\mathcal{H}$上稠密自伴半有界算子$A$的Friedrichs扩张的一个简化构造. \emph{symmetric pair}由密集定义的运算符$J: \mathcal{H}_1 \to \mathcal{H}_2$和$K: \mathcal{H}_2 \to \mathcal{H}_1$组成，它们在某种意义上是兼容的。利用$A$和$\mathcal{H}$对$\mathcal{H}_1$和$J$进行了适当的定义，证明了$(JJ^\star)^{-1}$是$A$的Friedrich推广.此外，我们使用相关的想法（包括无限包容的概念），以构建一个推广的建设的克莱因扩展$A$的作者在以前的论文中所列出的。这些结果被应用到无限网络上的图拉普拉斯算子的研究中，与希尔伯特空间$\ell^2(G)$和$\mathcal{H}_{\mathcal E}$（能量空间）有关。
Truncated moment problems for $J$-self-adjoint, $J$-skew-self-adjoint and $J$-unitary operators
本文研究了$J$ -自伴、$J$ -斜自伴和$J$ -酉算子的截断矩问题。给出了可解的条件。构造了矩量问题的一些正则解。作为副产品，得到了$J$ -反对称算子和$J$ -等距算子的一些推广结果.
Direct sums of trace maps and self-adjoint extensions
我们给出了一个简单的准则，使得一个可数的无限迹（赋值）映射的直和是迹映射。应用理论的自伴扩展的对称算子的直和提供，这给了一个替代的方法最近获得的结果Malamud Neidhardt和Kostenko Malamud使用正则化的直接和边界三元组。
Essential Self-Adjointness of Anti-Commutative Operators
本文研究了满足反对易关系的对称算子的自伴扩张。证明了Glimm-Jaffe-Nelson换位子定理的一个反交换型。它的应用抽象狄拉克算子也被认为是。
Boundary systems and (skew-)self-adjoint operators on infinite metric graphs
本文将拉格朗日子空间的概念推广到关于（斜）对称形式的自正交子空间，从而用自正交子空间刻画（斜）自伴算子和酉算子.通过保正交映射，这些刻画可以转移到（斜）对称算子的抽象边值空间。引入边界系统的概念，然后我们提出了一个统一的处理不同版本的边界三元组和相关概念的文献中处理。抽象结果的应用产生的描述的所有（斜）自伴实现的拉普拉斯和一阶导数算子的图。
In this note, we mainly investigate the validity of the identities $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$, where $T$ is a densely defined closable (or symmetric) operator.
Let $Ω$ be a domain in $\Ri^d$ with boundary $Γ$${\!,}$ $d_Γ$ the Euclidean distance to the boundary and $H=-\divv(C\,\nabla)$ an elliptic operator with $C=(\,c_{kl}\,)>0$ where $c_{kl}=c_{lk}$ are real, bounded, Lipschitz functions. We assume that $C\sim c\,d_Γ^{\,δ}$ as $d_Γ\to0$ in the sense of asymptotic analysis where $c$ is a strictly positive, bounded, Lipschitz function and $δ\geq0$. We also assume that there is an $r>0$ and a $ b_{δ,r}>0$ such that the weighted Hardy inequality \[ \int_{Γ_{\!\!r}} d_Γ^{\,δ}\,|\nabla ψ|^2\geq b_{δ,r}^{\,2}\int_{Γ_{\!\!r}} d_Γ^{\,δ-2}\,| ψ|^2 \] is vali…
We construct self-adjoint Laplacians and symmetric Markov semigroups on hyperbolic attractors, endowed with Gibbs $u$-measures. If the measure has full support, we can also conclude the existence of an associated symmetric diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on negatively curved manifolds the Laplacians we consider are self-adjoint extensions of well-known classical leafwise Laplacians. We observe a quasi-invariance property of energy densities in the $u$-conformal case and the existence of nonconstant functions of zero energ…
In this article, we prove and disprove several generalizations of unbounded versions of the Fuglede-Putnam theorem. As applications, we give conditions guaranteeing the commutativity of a bounded self-adjoint operator with an unbounded closed symmetric operator.
In a real Hilbert spaces H a smooth operator F is studied, whose derivative at each point of its domain is a symmetric operator. In terms of abstract boundary conditions locally self-adjoint extensions of this operator are described. We use some concepts and facts from symplectic differential geometry.
Boundary triples and Weyl functions for singular perturbations of self-adjoint operators
通过将自伴算子$A$限制为$N$（一个关于图范数闭的线性稠密集）而得到的对称算子$A_N$，我们确定了伴随算子$A_N^*$和相应Weyl函数的一个方便的边界三元组.这些对象为我们提供了$A_N$的自伴扩张及其预解式。
On J-Self-Adjoint Operators with Stable C-Symmetry
本文致力于发展Krein空间中的自伴算子（J-自伴算子）理论，其中包括由于C-对称的存在而引起的一些附加性质。主要讨论了最近关于对称算子S的J-自伴扩张的稳定C-对称性的概念。通过详细研究S具有亏损数<2，2>的情形，进一步特殊化了一般结果.
The aim of this paper is to develop an approach to obtain self-adjoint extensions of symmetric operators acting on anti-dual pairs. The main advantage of such a result is that it can be applied for structures not carrying a Hilbert space structure or a normable topology. In fact, we will show how hermitian extensions of linear functionals of involutive algebras can be governed by means of their induced operators. As an operator theoretic application, we provide a direct generalization of Parrott's theorem on contractive completion of $2$ by $2$ block operator-valued matrices. To exhibit the ap…
On determining the domain of the adjoint operator
给出了一个有助于计算伴随算子的定义域的定理。它可以用作例如对称算子的自伴性、形式正规算子的正规性或$H$-对称算子的$H$-自伴性的准则。以微分算子和无穷矩阵所给出的算子为例。
Maps on classes of Hilbert space operators preserving measure of commutativity
本文首先对L. Molnár和W.蒂默曼也就是说，我们将描述自伴随矩阵集合上的线性（不一定是双射）变换，这些变换保持交换子的酉不变范数。之后，我们将刻画作用在二维复希尔伯特空间上的自伴秩一投影集上的映射（不一定是线性或双射），这些映射使后一个量不变。最后，这一结果将被应用，以获得这样的双射的描述上的酉群和一组密度算子。
Extensions of symmetric operators I: The inner characteristic function case
给定希尔伯特空间上的对称线性变换，一个自然要考虑的问题是它的对称扩张集的特征。这个问题等价于研究一个固定的部分等距的部分等距扩张。通过构造自伴扩张集与所有大于或等于$Θ_B$的压缩解析函数$Φ$的集合之间的自然双射，给出了对称线性变换$B$的所有自伴扩张集的一个新的函数论刻画，该自伴扩张集具有等指数和内Livsic特征函数$Θ_B$ .此外，我们还刻画了在$Θ_B$是内部的情况下，$B$的所有具有相等指标的对称扩张$B'$的集合。
We prove essential self-adjointness of Dirac operators with Lorentz scalar potentials which grow sufficiently fast near the boundary $\partialΩ$ of the spatial domain $Ω\subset\mathbb R^d$. On the way, we first consider general symmetric first order differential systems, for which we identify a new, large class of potentials, called scalar potentials, ensuring essential self-adjointness. Furthermore, using the supersymmetric structure of the Dirac operator in the two dimensional case, we prove confinement of Dirac particles, i.e. essential self-adjointness of the operator, solely by magnetic f…
We investigate the spectrum of the generator of a self-adjoint transition semigroup of a (symmetric) Lévy process taking values in $d$--dimensional space.
Some Applications of Operator-Valued Herglotz Functions
我们考虑算子值Herglotz函数及其在自伴算子的自伴扰动和稠定闭对称算子的自伴扩张中的应用。我们的应用程序包括模型运营商的两种情况下，线性分数变换Herglotz运营商，弗里德里希和克莱因扩展的结果，实现定理类Herglotz运营商。此外，我们还研究了半直线上薛定谔算子的具体情形，并给出了Livsic [44]关于具有亏指数（1，1）的稠密对称算子的特殊情形下的拟厄米扩张的结果的两个例子.
On the reduction of powers of self-adjoint operators
本文研究自伴算子幂的约化问题，旨在建立使算子自伴的条件，特别关注$T^3=0$时算子复对称性的蕴含关系。
A linear operator on a Hilbert space $\mathbb{H}$, in the classical approach of von Neumann, must be symmetric to guarantee self-adjointness. However, it can be shown that the symmetry could be ommited by using a criterion for the graph of the operator and the adjoint of the graph. Namely, S is shown to be densely defined and closed if and only if $\{k + l : \{k, l\} \in G(S) \cap G(S)^*\} = \mathbb{H}$. In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear re…
We construct self-adjoint Laplacians and symmetric Markov semigroups on partially hyperbolic attractors and on hyperbolic attractors with singularities, endowed with Gibbs u-measures. If the measure has full support, we can also guarantee the existence of an associated symmetric Hunt diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on manifolds of negative sectional curvature the Laplacians we consider are self-adjoint extensions of well-known classical leafwise Laplacians.
Self-adjoint extensions with compact resolvent
本文研究了在可分复Hilbert空间中，闭对称算子具有紧谱的自共轭扩展的充要条件，并给出了其参数化方法。
Let $\mathcal H$ be a complex infinite-dimensional separable Hilbert space, and let $\mathcal K(\mathcal H)$ be the $C^*$-algebra of compact linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a symmetric sequence space. If $\{μ(n,x)\}$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{μ(n,x)\}\in E\}$ with $\|x\|_{\mathcal C_E}=\|\{μ(n,x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. Let $\mathcal C_E^h=\{x\in\mathcal C_E : x=x^*\}$ be the real Banach subspace of self-adjoint operators in …
General Aspects of PT-Symmetric and P-Self-Adjoint Quantum Theory in a Krein Space
在我们以前的工作中，我们提出了一个PT-对称量子理论的数学框架，特别是构造了一个Krein空间，PT-对称算子自然会在其中起作用。在这项工作中，我们探索和讨论了Krein空间中定义的理论的各种一般结果和方面，不仅是谱性质和PT对称性破缺，而且还有几个问题，这些问题对于理论在物理上是可接受的至关重要，例如状态向量的时间演化，概率解释，不确定关系，经典-量子对应，完备性，基的存在性等等。特别是，我们证明了对于给定的实经典系统，我们总能构造出相应的PT对称量子系统，这表明Krein空间中的PT对称理论是另一种量子化方案，而不是传统的Hermitian量子化方案在Hilbert空间中的推广.我们提出了一个假设的运营商是一个物理观察的框架。
Let $Ω$ be a domain in $\Ri^d$ with boundary $Γ$ and let $d_Γ$ denote the Euclidean distance to $Γ$. Further let $H=-\divv(C\nabla)$ where $C=(\,c_{kl}\,)>0$ with $c_{kl}=c_{lk}$ are real, bounded, Lipschitz continuous functions and $D(H)=C_c^\infty(Ω)$. Assume also that there is a $δ\geq0$ such that $\|C/d_Γ^{\,δ}-aI\|\to 0$ as $d_Γ\to0$ with $δ\geq0$ where $a$ is a bounded Lipschitz continuous function with $a\geqμ>0$ on a boundary layer $Γ_{\!\!r}=\{x\inΩ: d_Γ(x)<r\}$. Finally we require $|(\divv C).(\nabla d_Γ)|d_Γ^{\,-δ+1}$ to be bounded on~$Γ_{\!\!r}$. Then we prove that if $Ω$ is a $C^2…
On the unitary equivalence of absolutely continuous parts of self-adjoint extensions
经典的Weyl-von Neumann定理指出，对于可分Hilbert空间$\mathfrak H$中的任何自伴算子$A$，存在（非唯一的）Hilbert Schmidt算子$C = C^*$，使得扰动算子$A+C$具有纯点谱。我们感兴趣的是，这个结果是否仍然有效的非添加剂扰动考虑自伴扩展的一个给定的稠密定义的对称算子$A$在$\mathfrak H$和固定的扩展$A_0 = A_0^*$。我们证明了对于广泛的一类对称算子，扩张$\widetilde A = {\widetilde A}^*$和$A_0$的绝对连续部分是酉等价的，只要它们的预解式差是一个紧算子.也就是说，我们表明，这是真的，只要外尔函数$M(\cdot)$的一对$\{A,A_0\}$承认有界的限制$M(t) := \wlim_{y\to+0}M(t+iy)$的a. e。$t \in \mathbb{R}$。这一结果适用于对称算子和Sturm-Liouville算子与算子潜力的直和。
The work is devoted to the study of the similarity of a correct restriction to some self-adjoint operator in the case when the minimal operator is symmetric. The resulting theorem was applied to the Sturm-Liouville operator and the Laplace operator. It is shown that the spectrum of a non self-adjoint singularly perturbed operator is real and the corresponding system of eigenvectors forms a Riesz basis.
Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators
本文将广义边界三元组及Weyl函数推广至算子伴随对，给出诱导非空预解集闭算子的边界参数准则，并应用于具紧致边界的Lipschitz域上带复Lp势的Schrödinger算子。
We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussed.
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。
In this article, we give conditions guaranteeing the commutativity of a bounded self-adjoint operator with an unbounded closed symmetric operator.
We give an explicit description of all minimal self-adjoint extensions of a densely defined, closed symmetric operator in a Hilbert space with deficiency indices $(1, 1)$.
Multiplicatively spectrum-preserving maps on $C^{*}$-algebras
本文刻画含单位元$C^*$-代数自伴元之间的乘法谱保持满射，证明其由Jordan同构与中心对称决定，并解答Molnár的问题。
Maps preserving the spectrum of certain products of operators
设H是复希尔伯特空间，B（H）和S（H）分别是H上所有有界算子和所有自伴算子的空间.给出了B（H）上和S（H）上保持某些算子乘积谱的映射的具体形式。
Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields on curved spacetime
本文对辛空间上的一对算子，它们是彼此关于辛形式的伴随算子，（也就是说，它们是辛伴随的），如果它们对于控制辛形式的辛空间上的某个标量积是有界的，那么它们相对于与最初给定的一个规范相关的单参数标量积族是有界的，其中之一就是“净化”。作为一个典型的例子，我们考虑一个标量场的全球双曲时空的克莱因-戈登方程;经典系统描述的辛空间和时间演化的辛同胚（这是辛伴随其逆）。自然标积是诱导经典能量范数的标积，应用上述结果，它的“纯化”在量子化系统的单粒子空间上诱导出与准自由Hadamard态的两点函数所给出的拓扑相一致的拓扑。这些发现将导致新的结果有关的结构的局部（冯诺依曼）可观代数表示的准自由Hadamard状态的Klein-Gordon场在一个任意的全球双曲时空，如局部明确性，本地primarity和Haag对偶（也分裂和III_1型性质）。一个简短的审查这个圈子的概念，以及阿达玛状态的属性，形成文章的一部分。
On the number of square integrable solutions and self-adjointness of symmetric first order systems of differential equations
本文主要研究一阶对称系统$$ Jf'+Bf=λ{\cal H} f $$在区间$I$上的形式亏度指标${\cal N}_{\pm}(I)$，其中$I=\mathbb{R}$或$I=\mathbb{R}_\pm.$这里$J,B,{cal H}$是$n\times n$矩阵值函数，并且Hamilton量${\cal H}\ge 0$可以处处奇异.我们得到了两个结果，这样的系统有最小的数字${\cal N}_\pm(\mathbb{R})=0$（分别）。${\cal N}_\pm(\mathbb{R}_\pm)=n$）及其极大性判据${\cal N}_{\pm}(\mathbb{R}_+)=2n.$给出了正则系统具有中间数${\cal N}_\pm(\mathbb{R}_+)$的一些条件.我们还得到了著名的Titchmarsh-Sears定理的推广二阶Sturm-Liouville型方程。这包含由于Lidskii和Krein作为特殊情况的结果。重要的是要注意，通常上述系统不会产生算子，而是产生希尔伯特空间中的对称线性关系。这些关系进行了详细的调查。作为一个副产品，我们获得了非常短的证明（推广）的主要结果的文件由科根和罗夫Beketov（Proc。Soc. 74（1974/75））以及正则系统的拟正则性的判据。这涵盖了卡茨-克莱因定理和一些结果从引用的文件Kogan和Rofe-Beketo…
We compute the deficiency spaces of operators of the form $H_A{\hat{\otimes}} I + I{\hat{\otimes}} H_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von Neumann's theory. The structure of the deficiency spaces for this case was asserted already by Ibort, Marmo and Pérez-Pardo, but only proven under the restriction of $H_B$ having discrete, non-degenerate spectrum.
Quasi boundary triples and semibounded self-adjoint extensions
利用拟边界三元组及其Weyl函数的抽象概念，研究了对称算子的半有界自伴扩张.主要目的是提供新的充分条件的参数在边界空间中诱导自伴实现，并与衰减的Weyl函数的估计的下界的频谱。抽象的结果说明了一致椭圆二阶偏微分方程的区域上的非紧边界。
We consider symmetric Jacobi operators with recurrence coefficients such that the corresponding difference equation is in the limit circle case. Equivalently, this means that the associated moment problem is indeterminate. Our main goal is to find a representation for the resolvents of self-adjoint realizations $J$ of such Jacobi operators. This representation implies the classical Nevanlinna formula for the Cauchy-Stieltjes transforms of the spectral measures of the operators $J$. We also efficiently describe domains of the operators $J$ in terms of boundary conditions at infinity.
Almost commuting functions of almost commuting self-adjoint operators
设$A$和$B$是几乎交换的（即$AB-BA\in\bS_1$）自伴算子.在Besov类$B_{\be,1}^1(\R^2)$中构造了$\f$的函数演算$\f\mapsto\f(A,B)$.该泛函演算是线性的，算子$\f(A,B)$和$ψ(A,B)$几乎可交换为$\f,\,ψ\in B_{\be,1}^1(\R^2)$，$\f(A,B)=u(A)v(B)$，只要$\f(s,t)=u(s)v(t)$，Helton-Howe迹公式成立.主要工具是三重算子积分。
This work deals with the functional model for a class of extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices. On the basis of these formulae, we are able to construct wave operators and derive a new representation for the scattering matrix for pairs of such extensions in both self-adjoint and non-self-adjoint situati…
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