Extremal extensions for the sum of nonnegative selfadjoint relations

The sum of two nonnegative selfadjoint relations (multi-valued operators) and is a nonnegative relation. The class of all extremal extensions of the sum is characterized as products of relations via an auxiliary Hilbert space associated with and . The so-called form sum extension of is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with and . Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on and .

     

The range of linear fractional maps on the unit ball

In 1996, C. Cowen and B. MacCluer studied a class of maps on that they called linear fractional maps. Using the tools of Krein spaces, it can be shown that a linear fractional map is a self-map of the ball if and only if an associated matrix is a multiple of a Krein contraction. In this paper, we extend this result by specifying this multiple in terms of eigenvalues and eigenvectors of this matrix, creating an easily verified condition in almost all cases. In the remaining cases, the best possible results depending on fixed point and boundary behavior are given.

     

A reproducing kernel space model for -functions

A new model for generalized Nevanlinna functions will be presented. It involves Bezoutians and companion operators associated with certain polynomials determined by the generalized zeros and poles of . The model is obtained by coupling two operator models and expressed by means of abstract boundary mappings and the corresponding Weyl functions.

     

Weyl families of transformed boundary pairs

Let $(\mathfrak{L},\mathrm{\Gamma })$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space $\mathfrak{H}$. Let $M_{\mathrm{\Gamma }}$ be the Weyl family corresponding to $(\mathfrak{L},\mathrm{\Gamma })$. We cope with two main topics. First, since $M_{\mathrm{\Gamma }}$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation $M_{\mathrm{\Gamma }}(z)$, for some $z\in \mathbb{C}\setminus \mathbb{R}$, becomes a nontrivial task. Regarding $M_{\mathrm{\Gamma }}(z)$ as the (Shmul'yan) transform of $zI$ induced by Γ, we give conditions for the equality in $\overline{M_{\mathrm{\Gamma }}(z)}\subseteq \overline{M_{\overline{\mathrm{\Gamma }}}(z)}$ to hold and we compute the adjoint $M_{\overline{\mathrm{\Gamma }}}{(z)}^{\ast }$. As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for $T^{+}$ is nonempty. Based on the criterion for the closeness of $M_{\mathrm{\Gamma }}(z)$, we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family $M_{\mathrm{\Gamma }}$ corresponding to a boundary relation Γ for $T^{+}$ is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$ with its Weyl family $M_{{\mathrm{\Gamma }}^{\prime }}$. The transformation scheme is either ${\mathrm{\Gamma }}^{\prime }=\mathrm{\Gamma }{V}^{-1}$ or ${\mathrm{\Gamma }}^{\prime }=V\mathrm{\Gamma }$ with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between $(\mathfrak{L},\mathrm{\Gamma })$ and $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$; the formula for $M_{{\mathrm{\Gamma }}^{\prime }}-M_{\mathrm{\Gamma }}$, for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple $(\mathfrak{L},{\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1)$ with $ker\mathrm{\Gamma }=T$ and $T_0=T_0^{\ast }$ (second scheme, Hilbert space case).     

A characterization of semibounded selfadjoint operators

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.

     

Nevanlinna Functions,Perturbation Formulas,and Triplets of Hilbert Spaces

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.     

Boundary value problems with eigenvalue depending boundary conditions

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 P_K (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)     

The Schur algorithm and coefficient characterizations for generalized Schur functions

In this paper we analyze the existence of a Schur algorithm and obtain coefficient characterizations for the functions in a generalized Schur class. An application to an interpolation problem of Carathéodory type raised by M.G. Krein and H. Langer is indicated.

     

Products of generalized Nevanlinna functions with symmetric rational functions

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.     

An example of unitary equivalence between self-adjoint extensions and their parameters

The spectral problem for self-adjoint extensions is studied using the machinery of boundary triplets. For a class of symmetric operators having Weyl functions of a special type we calculate explicitly the spectral projections in the form of operator-valued integrals. This allows one to give a constructive proof of the fact that, in certain intervals, the resulting self-adjoint extensions are unitarily equivalent to the parameterizing boundary operator acting in a smaller space, and one is able to provide an explicit form for the associated unitary transform. Applications to differential operators on metric graphs and to direct sums are discussed.     

非齐性广义Morrey空间上双线性强奇异Calder\''{o}n-Zygmund算子及交换子

本文的主要建立非齐性度量测度空间上双线性强奇异积分算子$\widetilde{T}$及交换子$\widetilde{T}_{b_{1},b_{2}}$在广义Morrey空间$M^{u}_{p}(\mu)$上的有界性. 在假设Lebesgue可测函数$u, u_{1}, u_{2}\in\mathbb{W}_{\tau}$, $u_{1}u_{2}=u$,且$\tau\in(0,2)$. 证明了算子$\widetilde{T}$是从乘积空间$M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$到空间$M^{u}_{p}(\mu)$有界的, 也是从乘积空间$M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$到广义弱Morrey空间$WM^{u}_{p}(\mu)$有界的,其中$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$及$1相似文献