Spectral Analysis of Certain Self-Adjoint Differential Operators in a Space with Indefinite Metric

A second-order differential operator self-adjoint with respect to an indefinite metric on the circle is considered. The spectral resolution of this operator is found. The hypergeometric function is used in the computation of the Plancherel measure.     

An addendum to the multiplication and factorization theorems of brodskiǐ—livšic—potapov

In this paper the classical Brodskiǐ—Liv?ic— operator colligation is generalized to certain pairs of bounded linear operators and the corresponding characteristic operator—valued (transfer) function is introduced. The fundamental results due to M.S. Brodskiǐ, M.S. Liv?ic, and V.P. Potapov are then extended to such colligations. These new types of colligations can be used to obtain, for instance, realization results for general Herglotz—Nevanlinna functions.     

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.     

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.

     

Rank One Perturbations in a Pontryagin Space with One Negative Square

Let N₁ denote the class of generalized Nevanlinna functions with one negative square and let N_1, 0 be the subclass of functions Q(z)∈N₁ with the additional properties lim_y→∞ Q(iy)/y=0 and lim sup_y→∞ 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 N_1, 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.     

Description of bilinear forms generated by indefinite vector measures

We describe a correlation function generated by a J-orthogonal indefinite measure with values in a Krein space.     

Solvability of Boundary Value Problems for Operator-Differential Equations of Mixed Type: The Degenerate Case

We obtain some results on solvability of boundary value problems for the equation B u _t –L u =f (t(0,)), where B and L are selfadjoint and dissipative operators defined in a Hilbert space E. The kernel of B may be nontrivial and L is uniformly dissipative on a subspace M of finite codimension.     

P耗散算子的研究

本文利用广义半内积空间及广义不定度规空间,解决Ｂａｎａｃｈ空间中Ｐ耗散算子的极大耗散扩张表示,并得到很有意义的应用．     

A characterization of extended monotone metrics

Classical information geometry has emerged from the study of geometrical aspect of the statistical estimation. Cencov characterized the Fisher metric as a canonical metric on probability simplexes from a statistical point of view, and Campbell extended the characterization of the Fisher metric from probability simplexes to positive cone . In quantum information geometry, quantum states which are represented by positive Hermitian matrices with trace one are regarded as an extension of probability distributions. A quantum version of the Fisher metric is introduced, and is called a monotone metric. Petz characterized the monotone metrics on the space of all quantum states in terms of operator monotone functions. A purpose of the present paper is to extend a characterization of monotone metrics from the space of all states to the space of all positive Hermitian matrices on finite dimensional Hilbert space. This characterization corresponds quantum modification of Campbell’s work.     

Schrodinger算子的极大耗散扩张

本文研究Ｓｃｈｒｏｄｉｎｇｅｒ算子的极大耗散扩张，引入广义不定度规空间，得到在自然边界空间中，Ｓｃｈｒｏｄｉｎｇｅｒ算子极大耗散扩张表示形式，为进一步研究非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程无穷维动力系统的长期混沌行为作准备。     

Semiconstant measures on hyperbolic logics

We characterize the set of all semiconstant measures on the hyperbolic logics of projections in indefinite metric spaces and describe the set of all probability measures on these logics.

     

Zeros of nonpositive type of generalized Nevanlinna functions with one negative square

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.     

Geometry of ℑ-Spaces and Properties of Reversing Operators

Mathematical Notes -     

Minimal and maximal operator spaces and operator systems in entanglement theory

We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.     

Dirac structures and their composition on Hilbert spaces

Dirac structures appear naturally in the study of certain classes of physical models described by partial differential equations and they can be regarded as the underlying power conserving structures. We study these structures and their properties from an operator-theoretic point of view. In particular, we find necessary and sufficient conditions for the composition of two Dirac structures to be a Dirac structure and we show that they can be seen as Lagrangian (hyper-maximal neutral) subspaces of Kre?n spaces. Moreover, special emphasis is laid on Dirac structures associated with operator colligations. It turns out that this class of Dirac structures is linked to boundary triplets and that this class is closed under composition.     

Loewner's theorem for kernels having a finite number of negative squares

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.

     

Norms and spectral radii of composition operators acting on the Dirichlet space

We obtain new upper bounds on the norms of univalently induced composition operators acting on the Dirichlet space and compute explicitly the norms for univalent symbols whose range is the disk minus a set of measure zero. As an application, we show that the spectral radius of every univalently induced composition operator on the Dirichlet space is equal to one.