Generalized Nevanlinna Functions Research Articles

Characterization of Weyl Functions in the Class of Operator-Valued Generalized Nevanlinna Functions

Characterization of Weyl Functions in the Class of Operator-Valued Generalized Nevanlinna Functions

Reducibility of Self-Adjoint Linear Relations and Application to Generalized Nevanlinna Functions

Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function $Q$, represented by a self-adjoint linear relation $A$, is decomposed by means of the reducing subspaces of $A$. The sum of two functions $Q_{i}{\in N}_{\kappa_{i}}\left( \mathcal{H} \right),\thinspace i=1,\thinspace 2$, minimally represented by the triplets $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$, is also studied. For that purpose, a model $( \tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma } )$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$ is created. By means of that model, necessary and sufficient conditions for $\kappa =\kappa_{1}+\kappa_{2}$ are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation $A$ affect reducing subspaces of $A$ and decomposition of the corresponding function $Q$.

Связь обратного преобразования Шура обобщённого класса Неванлинны с рациональными матричными функциями специального вида

In this paper we consider classical Schur transformation and inverse Schur transformation for generalized Nevanlinna functions.

Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions

UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function Q represented by a self-adjoint linear relation A in a Pontryagin space is decomposed by means of the reducing subspaces of A . The sum of two functions Q i ∈ N κ i ( ℋ ) , i = 1,2 , minimally represented by the triplets ( 𝒦 i , A i , Γ i ) is also studied. For this purpose, we create a model ( 𝒦 ˜ , A ˜ , Γ ˜ ) to represent Q : = Q 1 + Q 2 in terms of ( 𝒦 i , A i , Γ i ) . By using this model, necessary and sufficient conditions for κ = κ 1 + κ 2 are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation A affect the reducing subspaces of A and the decomposition of the corresponding function Q .

Characterizations of Herglotz\u2013Nevanlinna Functions Using Positive Semi-Definite Functions and the Nevanlinna Kernel in Several Variables

In this paper, we give several characterizations of Herglotz–Nevanlinna functions in terms of a specific type of positive semi-definite functions called Poisson-type functions. This allows us to propose a multidimensional analogue of the classical Nevanlinna kernel and a definition of generalized Nevanlinna functions in several variables. Furthermore, a characterization of the symmetric extension of a Herglotz–Nevanlinna function is also given. The subclass of Loewner functions is discussed as well, along with an interpretation of the main result in terms of holomorphic functions on the unit polydisk with non-negative real part.

Generalized Schur Functions as Multivalent Functions

The multivalency approach to generalized Nevanlinna functions established in Wietsma (Indag Math 29:997–1008, 2018) is here extended to the related class of generalized Schur functions giving thereby rise to new characterizations for this class of functions as well as a straightforward function-theoretical proof of its factorization. In particular, this multivalency approach explains how the well-known factorizations of the two mentioned classes of functions differ from each other. Indeed, by this approach a new factorization of generalized Schur functions is obtained which is more directly connected to the factorization of generalized Nevanlinna functions. These results demonstrate that multivalency is a valuable concept for the complete understanding of the mentioned classes of functions.

Generalized Schur\u2013Nevanlinna functions and their realizations

Pontryagin space operator valued generalized Schur functions and generalized Nevanlinna functions are investigated by using discrete-time systems, or operator colligations, and state space realizations. It is shown that generalized Schur functions have strong radial limit values almost everywhere on the unit circle. These limit values are contractive with respect to the indefinite inner product, which allows one to generalize the notion of an inner function to Pontryagin space operator valued setting. Transfer functions of self-adjoint systems such that their state spaces are Pontryagin spaces, are generalized Nevanlinna functions, and symmetric generalized Schur functions can be realized as transfer functions of self-adjoint systems with Kreĭn spaces as state spaces. A criterion when a symmetric generalized Schur function is also a generalized Nevanlinna function is given. The criterion involves the negative index of the weak similarity mapping between an optimal minimal realization and its dual. In the special case corresponding to the generalization of an inner function, a concrete model for the weak similarity mapping can be obtained by using the canonical realizations.

Arov–Krein Entropy Functionals and Indefinite Interpolation Problems

We generalize the notion of the Arov-Krein entropy functional for the case of generalized Nevanlinna functions and obtain a representation of these functionals on solutions of indefinite interpolation problems. The case of indefinite Caratheodory problem and application to Szeg\H{o} limit formula for this nonclassical case are considered in greater detail.

The factorization of generalized Nevanlinna functions and the invariant subspace property

The well-known invariant subspace property of selfadjoint relations (multi-valued operators) in Pontryagin spaces is shown to be equivalent to the factorization property of (scalar) generalized Nevanlinna functions. This connection is established by a new realization for generalized Nevanlinna functions explicitly reflecting this connection. Combining this result with the new function-theoretic proof for the factorization property of generalized Nevanlinna functions contained in Wietsma (2018) immediately yields a new proof for the invariant subspace property of selfadjoint relations in Pontryagin spaces.

Generalized Nevanlinna functions and multivalency

A new characterization of generalized Nevanlinna functions in terms of multivalency is established. By analyzing multivalent functions additional novel characterizations of (generalized) Nevanlinna functions are obtained. As a particular consequence of these investigations a function-theoretic proof of the factorization property of generalized Nevanlinna functions is obtained.

Minimal realizations of generalized Nevanlinna functions

Minimal realizations of generalized Nevanlinna functions that carry the information on their generalized poles of nonpositive type in an explicit form are established. These realizations are based on a modification of the basic canonical factorization of generalized Nevanlinna functions whereby the non-minimality problems in realizations that are based directly on the canonical factorization are circumvented.

Distributional representations of Nκ(∞)‐functions

The subclasses of the classes of generalized Nevanlinna functions appear in the context of Pontryagin space models, where they correspond to model relations having a particular spectral behaviour. Applications are found, for instance, in the investigation of differential expressions with singular coefficients. We study representations of ‐functions as Cauchy‐type integrals in a distributional sense and characterize the class of distributions occurring in such representations. We make explicit how the Pontryagin space model of an ‐function is related to the multiplication operator in the L2‐space of the measure which describes the action of the representing distribution away from infinity. Moreover, we determine the distributional representations of a pair of functions associated with a symmetric generalized Nevanlinna function.

Analytic characterizations of Jordan chains by pole cancellation functions of higher order

In this paper the analytic characterization of generalized poles of operator valued generalized Nevanlinna functions (including the length of Jordan chains of the representing relation) is completed. In particular, given a Jordan chain of the representing relation of length ℓ, we show that there exists a pole cancellation function of order at least ℓ, and, moreover, the construction shows that it is of surprisingly simple form.

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.

Global and local behavior of zeros of nonpositive type

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 is being studied. In particular, it is shown that it is continuous and its behavior in the points where the function extends through the real line is investigated.

Some Propositions on Generalized Nevanlinna Functions of the ClassNk

Some propositions on the generalized Nevanlinna functions are derived. We indicate mainly that (1) the negative inertia index of a Hermitian generalized Loewner matrix generated by a generalized Nevanlinna function in the classNκdoes not exceedκ. This leads to an equivalent definition of a generalized Nevanlinna function; (2) if a generalized Nevanlinna function in the classNκhas a uniform asymptotic expansion at a real pointαor at infinity, then the negative inertia index of the Hankel matrix constructed with the partial coefficients of that asymptotic expansion does not exceedκ. Also, an explicit formula for the negative index of a real rational function is given by using relations among Loewner, Bézout, and Hankel matrices. These results will provide first tools for the solution of the indefinite truncated moment problems together with the multiple Nevanlinna-Pick interpolation problems in the classNκbased on the so-called Hankel vector approach.

On the negative squares of a class of self‐adjoint extensions in Krein spaces

AbstractA description of all exit space extensions with finitely many negative squares of a symmetric operator of defect one is given via Krein's formula. As one of the main results an exact characterization of the number of negative squares in terms of a fixed canonical extension and the behaviour of a function τ (that determines the exit space extension in Krein's formula) at zero and at infinity is obtained. To this end the class of matrix valued \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {D}_\kappa ^{n\times n}$\end{document}‐functions is introduced and, in particular, the properties of the inverse of a certain \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {D}_\kappa ^{2\times 2}$\end{document}‐function which is closely connected with the spectral properties of the exit space extensions with finitely many negative squares is investigated in detail. Among the main tools here are the analytic characterization of the degree of non‐positivity of generalized poles of matrix valued generalized Nevanlinna functions and some extensions of recent factorization results.

Indefinite Hamiltonian systems whose Titchmarsh-Weyl coefficients have no finite generalized poles of non-positive type

The two-dimensional Hamiltonian system (*) y'(x)=zJH(x)y(x), x∈(a,b), where the Hamiltonian H takes non-negative 2x2-matrices as values, and $J:= \begin{pmatrix} 0 & -1 1 & 0 \end{pmatrix}$, has attracted a lot of interest over the past decades. Special emphasis has been put on operator models and direct and inverse spectral theorems. Weyl theory plays a prominent role in the spectral theory of the equation, relating the class of all equations (*) to the class N0 of all Nevanlinna functions via the construction of Titchmarsh–Weyl coefficients. In connection with the study of singular potentials, an indefinite (Pontryagin space) analogue of equation (*) was proposed, where the 'general Hamiltonian' is allowed to have a finite number of inner singularities. Direct and inverse spectral theorems, relating the class of all general Hamiltonians to the class <N∞ of all generalized Nevanlinna functions, were established. In the present paper, we investigate the spectral theory of general Hamiltonians having a particular form, namely, such which have only one singularity and the interval to the left of this singularity is a so-called indivisible interval. Our results can comprehensively be formulated as follows. — We prove direct and inverse spectral theorems for this class, i.e. we establish an intrinsic characterization of the totality of all Titchmarsh–Weyl coefficients corresponding to general Hamiltonians of the considered form. — We determine the asymptotic growth of the fundamental solution when approaching the singularity. — We show that each solution of the equation has 'polynomially regularized' boundary values at the singularity. Besides the intrinsic interest and depth of the presented results, our motivation is drawn from forthcoming applications: the present theorems form the core for our study of Sturm–Liouville equations with two singular endpoints and our further study of the structure theory of general Hamiltonians (both to be presented elsewhere).

A Schur transformation for functions in a general class of domains

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.

Truncated moment problems in the class of generalized Nevanlinna functions

AbstractTruncated moment and related interpolation problems in the class of generalized Nevanlinna functions are investigated. General solvability criteria will be established and complete parametrizations of solutions are given. The framework used in the paper allows the treatment of even and odd order problems in a parallel manner. The main new results concern the case where the corresponding Hankel matrix of moments is degenerate. One of the new effects in the indefinite case is that the degenerate moment problem may have infinitely many solutions. However, with a careful application of an indefinite analogue of a step‐by‐step Schur algorithm a complete description of the set of solutions will be obtained.