共查询到20条相似文献，搜索用时 15 毫秒
1.
Let N1 denote the class of generalized Nevanlinna functions with one negative square and let N1, 0 be the subclass of functions Q(z)∈N1 with the additional properties limy→∞ Q(iy)/y=0 and lim supy→∞ y |Im Q(iy)|<∞. These classes form an analytic framework for studying (generalized) rank one perturbations A(τ)=A+τ[·, ω] ω in a Pontryagin space setting. Many functions appearing in quantum mechanical models of point interactions either belong to the subclass N1, 0 or can be associated with the corresponding generalized Friedrichs extension. In this paper a spectral theoretical analysis of the perturbations A(τ) and the associated Friedrichs extension is carried out. Many results, such as the explicit characterizations for the critical eigenvalues of the perturbations A(τ), are based on a recent factorization result for generalized Nevanlinna functions. 相似文献
2.
New classes of generalized Nevanlinna functions, which under multiplication with an arbitrary fixed symmetric rational function remain generalized Nevanlinna functions, are introduced. Characterizations for these classes of functions are established by connecting the canonical factorizations of the product function and the original generalized Nevanlinna function in a constructive manner. Also, a detailed functional analytic treatment of these classes of functions is carried out by investigating the connection between the realizations of the product function and the original function. The operator theoretic treatment of these realizations is based on the notions of rigged spaces, boundary triplets, and associated Weyl functions. 相似文献
3.
Vladimir Derkach Seppo Hassi Mark Malamud Henk de Snoo 《Transactions of the American Mathematical Society》2006,358(12):5351-5400
The concepts of boundary relations and the corresponding Weyl families are introduced. Let be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space , let be an auxiliary Hilbert space, let and let be defined analogously. A unitary relation from the Krein space to the Krein space is called a boundary relation for the adjoint if . The corresponding Weyl family is defined as the family of images of the defect subspaces , , under . Here need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space and the class of unitary relations , it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every -valued maximal dissipative (for ) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.
4.
Let S be a closed symmetric operator with defect numbers (1,1) in a Hilbert space ?? and let A be a selfadjoint operator extension of S in ??. Then S is necessarily a graph restriction of A and the selfadjoint extensions of S can be considered as graph perturbations of A, cf. [8]. Only when S is not densely defined and, in particular, when S is bounded, 5 is given by a domain restriction of A and the graph perturbations reduce to rank one perturbations in the sense of [23]. This happens precisely when the Q - function of S and A belongs to the subclass No of Nevanlinna functions. In this paper we show that by going beyond the Hilbert space ?? the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space ?? is given a one-dimensional extension, or the use of Hilbert space triplets associated with A is invoked. If the Q - function of S and A belongs to the subclass N1 of Nevanlinna functions, then it is convenient to describe the selfadjoint extensions of S including its generalized Friedrichs extension (see [6]) by interpolating the original triplet, cf. [5]. For the case when A is semibounded, see also [4]. We prove some invariance properties, which imply that such an interpolation is independent of the (nonexceptional) extension. 相似文献
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6.
Seppo Hassi Michael Kaltenbä ck Henk de Snoo 《Proceedings of the American Mathematical Society》1997,125(9):2681-2692
For a class of closed symmetric operators with defect numbers it is possible to define a generalization of the Friedrichs extension, which coincides with the usual Friedrichs extension when is semibounded. In this paper we provide an operator-theoretic interpretation of this class of symmetric operators. Moreover, we prove that a selfadjoint operator is semibounded if and only if each one-dimensional restriction of has a generalized Friedrichs extension.
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8.
In this paper we study a class of generalized Fock spaces associated with the Dunkl operator. Next we introduce the commutator relations between the Dunkl operator and multiplication operator which leads to a generalized class of Weyl relations for the Dunkl kernel. 相似文献
9.
By a theorem of Loewner, a continuously differentiable real-valued function on a real interval whose difference quotient is a nonnegative kernel is the restriction of a holomorphic function which has nonnegative imaginary part in the upper half-plane and is holomorphic across the interval. An analogous result is obtained when the difference-quotient kernel has a finite number of negative squares.
10.
讨论了W12[a,b]能否扩大为含有有间断点函数的再生核空间的问题.结论是:若再生核空间W W12[a,b]含有有间断点的函数,则间断点必固定、间断点个数必有限且非端点a,b.进一步,我们构造了函数含有n个间断点的再生核空间并给出其再生核表达式. 相似文献
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12.
In this paper we present a framework in which the Schur transformation and the basic interpolation problem for generalized Schur functions, generalized Nevanlinna functions and the like can be studied in a unified way. The basic object is a general class of functions for which a certain kernel has a finite number of negative squares. The results are based on and generalize those in previous papers of the first three authors on the Schur transformation in an indefinite setting. 相似文献
13.
Zoltá n Sebestyé n Jan Stochel 《Proceedings of the American Mathematical Society》2007,135(5):1389-1397
The set of all positive selfadjoint extensions of a positive operator (which is not assumed to be densely defined) is described with the help of the partial order which is relevant to the theory of quadratic forms. This enables us to improve and extend a result of M. G. Krein to the case of not necessarily densely defined operators .
14.
Georg Schneider 《Proceedings of the American Mathematical Society》2004,132(8):2399-2409
We consider Hankel operators of the form . Here . We show that in the case of one complex dimension the Hankel operators are compact but not Hilbert-Schmidt if 2k$\">.
15.
Jussi Behrndt 《Mathematische Nachrichten》2009,282(5):659-689
We investigate some classes of eigenvalue dependent boundary value problems of the form where A ? A+ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ0, Γ1 are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent PK (à – λ)–1|K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
16.
Summary A bounded, not necessarily everywhere defined, nonnegative operator A in a Hilbert space <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource
Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathfrak{H}$ is assumed
to intertwine in a certain sense two bounded everywhere defined operators B and C. If the range of A is provided with a natural inner product then the operators B and C induce two new operators on the completion space. This construction is used to show the existence of selfadjoint and nonnegative
extensions of B*A and C*A. 相似文献
17.
本文利用微分算子插值样条函数的方法给出了W12[a,b]空间再生核构造的新方法,证实了求解微分方程边值问题的方法([1]再生核空间数值分析[M].科学出版社,2004),其实是本文方法的一种特例. 相似文献
18.
本文研究了Bergman空间上的复合算子的范数与再生核的关系,证明了紧复合算子C的范数‖C‖=sup{‖C*kw‖:w∈D}的充要条件是(0)=0或是仿射映射,即(z)=sz+t,s,t是满足|s|+|t|<1的常数,其中kw为Bergman空间的规范再生核, C*是C的共轭算子. 相似文献
19.
Henk de Snoo 《Journal of Mathematical Analysis and Applications》2011,382(1):399-417
A generalized Nevanlinna function Q(z) with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by Qτ(z)=(Q(z)−τ)/(1+τQ(z)), τ∈R∪{∞}, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ) as a function of τ defines a path in the closed upper halfplane. Various properties of this path are studied in detail. 相似文献
20.
W_2~m[a，b]空间中再生核的计算(Ⅰ) 总被引：1，自引：1，他引：0
Zhang Xinjian Long Han 《计算数学》2008,(3)
本文用Green函数与伴随函数方法讨论由一般线性微分算子确定的再生核的具体计算.提出了基本Green函数与基本再生核的概念,它们是由微分算子和初值点唯一确定的;指出基本再生核的计算可转化为求解微分方程的初值问题,一般的再生核可由基本再生核的投影而得到;最后用例子说明了所给方法. 相似文献