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For a pair of bounded linear Hilbert space operators A and B one considers the Lebesgue type decompositions of B with respect to A into an almost dominated part and a singular part, analogous to the Lebesgue decomposition for a pair of measures in which case one speaks of an absolutely continuous and a singular part. A complete parametrization of all Lebesgue type decompositions will be given, and the uniqueness of such decompositions will be characterized. In addition, it will be shown that the almost dominated part of B in a Lebesgue type decomposition has an abstract Radon–Nikodym derivative with respect to the operator A.

Columns and rows are operations for pairs of linear relations in Hilbert spaces, modelled on the corresponding notions of the componentwise sum and the usual sum of such pairs. The introduction of matrices whose entries are linear relations between underlying component spaces takes place via the row and column operations. The main purpose here is to offer an attempt to formalize the operational calculus for block matrices, whose entries are all linear relations. Each block relation generates a unique linear relation between the Cartesian products of initial and final Hilbert spaces that admits particular properties which will be characterized. Special attention is paid to the formal matrix multiplication of two blocks of linear relations and the connection to the usual product of the unique linear relations generated by them. In the present general setting these two products need not be connected to each other without some additional conditions.

With a closed symmetric operator A in a Hilbert space H a triple Π={H,Γ0,Γ1} of a Hilbert space H and two abstract trace operators Γ0 and Γ1 from A∗ to H is called a generalized boundary triple for A∗ if an abstract analogue of the second Green's formula holds. Various classes of generalized boundary triples are introduced and corresponding Weyl functions M(·) are investigated. The most important ones for applications are specific classes of boundary triples for which Green's second identity admits a certain maximality property which guarantees that the corresponding Weyl functions are Nevanlinna functions on H, i.e. M(·)∈R(H), or at least they belong to the class R∼(H) of Nevanlinna families on H. The boundary condition Γ0f=0 determines a reference operator A0(=kerΓ0). The case where A0 is selfadjoint implies a relatively simple analysis, as the joint domain of the trace mappings Γ0 and Γ1 admits a von Neumann type decomposition via A0 and the defect subspaces of A. The case where A0 is only essentially selfadjoint is more involved, but appears to be of great importance, for instance, in applications to boundary value problems e.g. in PDE setting or when modeling differential operators with point interactions. Various classes of generalized boundary triples will be characterized in purely analytic terms via the Weyl function M(·) and close interconnections between different classes of boundary triples and the corresponding transformed/renormalized Weyl functions are investigated. These characterizations involve solving direct and inverse problems for specific classes of operator functions M(·). Most involved ones concern operator functions M(·)∈R(H) for which τM(λ)(f,g)=(2iImλ)−1[(M(λ)f,g)−(f,M(λ)g)],f,g∈domM(λ),defines a closable nonnegative form on H. It turns out that closability of τM(λ)(f,g) does not depend on λ∈C± and, moreover, that the closure then is a form domain invariant holomorphic function on C± while τM(λ)(f,g) itself need not be domain invariant. In this study we also derive several additional new results, for instance, Kreĭn‐type resolvent formulas are extended to the most general setting of unitary and isometric boundary triples appearing in the present work. In part II of the present work all the main results are shown to have applications in the study of ordinary and partial differential operators.

An isometric operator V in a Pontryagin space H is called standard, if its domain and the range are nondegenerate subspaces in H. A description of coresolvents for standard isometric operators is known and basic underlying concepts that appear in the literature are unitary colligations and characteristic functions. In the present paper generalized coresolvents of non-standard Pontryagin space isometric operators are described. The methods used in this paper rely on a new general notion of boundary pairs introduced for isometric operators in a Pontryagin space setting. Even in the Hilbert space case this notion generalizes the earlier concept of boundary triples for isometric operators and offers an alternative approach to study operator valued Schur functions without any additional invertibility requirements appearing in the ordinary boundary triple approach., 42 pages

Hain-Lust equations appear in magnetohydrodynamics. They are Sturm-Liouville equations with coefficients depending rationally on the eigenvalue parameter. In this paper such equations are connected with a 2 x 2 system of differential equations, where the dependence on the eigenvalue parameter is linear. By means of this connection Weyl's fundamental limit-point/limit-circle classification is extended to a general setting of Hain-Lust-type equations.

The extremal maximal sectorial extensions of a not necessarily densely defined sectorial relation (multivalued linear operator) in a Hilbert space are characterized in terms of a construction which goes back to Sebestyen and Stochel. In particular the two extreme maximal sectorial extensions, namely the Friedrichs extension and the Krein extension, are characterized. For this purpose a survey is given of the connection between closed sectorial forms and maximal sectorial relations. (C) 2017 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

A linear relation from one Hilbert space to another Hilbert space is a linear subspace of the product of these spaces. In this chapter some material about such linear relations is presented and it is shown how linear operators, whether densely defined or not, fit in this context.

Second-order Sturm-Liouville differential expressions generate self-adjoint differential operators in weighted L2-spaces on an interval (a, b).

For the multi-dimensional Schrodinger operator -∆+V with a bounded real potential V on a bounded domain \( \varOmega \subset {\mathbb{R}}^{n} \) with a C2-smooth boundary a boundary triplet and a Weyl function will be constructed.

The classes of Weyl functions and more generally of Nevanlinna functions will be studied from the point of view of reproducing kernel Hilbert spaces.

Semibounded relations in a Hilbert space automatically have equal defect numbers, so that there are always self-adjoint extensions. In this chapter the semibounded self-adjoint extensions of a semibounded relation will be investigated.

In this chapter the spectrum of a self-adjoint operator or relation will be completely characterized in terms of the analytic behavior and the limit properties of the Weyl function.

The basic properties of boundary triplets for closed symmetric operators or relations in Hilbert spaces are presented. These triplets give rise to a parametrization of the intermediate extensions of symmetric relations, in particular of the self-adjoint extensions.

This monograph presents a comprehensive survey of modern operator techniques for boundary value problems and spectral theory. Included are self-contained treatments of the extension theory of symmetric operators and relations, complete spectral characterizations of self-adjoint operators in terms of the analytic properties of Weyl functions, form methods for semibounded operators, and functional analytic models in reproducing kernel Hilbert spaces. These abstract methods are illustrated for various applications, involving Sturm-Liouville operators, canonical systems of differential equations, and multidimensional Schrödinger operators., In dieser Monographie werden moderne operatortheoretische Techniken zur Untersuchung von Randwert- und Spektralproblemen entwickelt. Es werden unter anderem die Erweiterungstheorie von symmetrischen Operatoren und Relationen, eine vollständige spektrale Beschreibung von selbstadjungierten Operatoren mittels analytischer Eigenschaften der Weylfunktionen, Formmethoden für halbbeschränkte Operatoren, und funktionalanalytische Modelle in Hilberträumen mit reproduzierenden Kern, diskutiert. Die abstrakte Theorie wird mit verschiedenen Anwendungsbeispielen, wie etwa Sturm-Liouville Operatoren, kanonische Differentialgleichungssysteme, und multidimensionale Schrödinger Operatoren, illustriert

Boundary value problems for regular and singular canonical systems of differential equations are investigated.

Selfadjoint relations in Pontryagin spaces do not possess a spectral family completely characterizing them in the way that selfadjoint relations in Hilbert spaces do. Here it is shown that a combination of a factorization of generalized Nevanlinna functions with the standard spectral family of selfadjoint relations in Hilbert spaces can function as a spectral family for selfadjoint relations in Pontryagin spaces. By this technique additive decompositions are established for generalized Nevanlinna functions and selfadjoint relations in Pontryagin spaces.

Let T be a closed linear relation from a Hilbert space \({\mathfrak H}\) to a Hilbert space \({\mathfrak K}\) and let \(B \in \mathbf {B}({\mathfrak K})\) be selfadjoint. It will be shown that the relation T∗(I + iB)T is maximal sectorial via a matrix decomposition of B with respect to the orthogonal decomposition \({\mathfrak H}=\mathrm {d}\overline {\mathrm {om}}\, T^* \oplus \mathrm {mul}\, T\). This leads to an explicit expression of the corresponding closed sectorial form. These results include the case where mul T is invariant under B. The more general description makes it possible to give an expression for the extremal maximal sectorial extensions of the sum of sectorial relations. In particular, one can characterize when the form sum extension is extremal.

In this paper we consider sectorial operators, or more generally, sectorial relations and their maximal-sectorial extensions in a Hilbert space ${\mathfrak{H}}$ . Our particular interest is in sectorial relations $S$ , which can be expressed in the factorized form \begin{equation*}S=T^{*}(I+iB)T\qquad \text{or}\qquad S=T(I+iB)T^{*},\end{equation*} where $B$ is a bounded self-adjoint operator in a Hilbert space ${\mathfrak{K}}$ and $T:{\mathfrak{H}}\to {\mathfrak{K}}$ (or $T:{\mathfrak{K}}\to {\mathfrak{H}}$ , respectively) is a linear operator or a linear relation which is not assumed to be closed. Using the specific factorized form of $S$ , a description of all the maximal-sectorial extensions of $S$ is given, along with a straightforward construction of the extreme extensions $S_{F}$ , the Friedrichs extension, and $S_{K}$ , the Kreĭn extension of $S$ , which uses the above factorized form of $S$ . As an application of this construction, we also treat the form sum of maximal-sectorial extensions of two sectorial relations.

Let M be a Hilbert space. In this paper we study a class \(\mathcal{RS}{\mathfrak(m)}\) of operator functions that are holomorphic in the domain \(\mathbb{C} \setminus \{(-\infty, -1] \ \cup \ [1, +\infty)\}\) and whose values are bounded linear operators in \(\mathfrak{m}\). The functions in \(\mathcal{RS}{\mathfrak(m)}\) are Schur functions in the open unit disk \(\mathbb{D}\) and, in addition, Nevanlinna functions in \(\mathbb{C}_{+} \cup \mathbb{C}_{-}\). Such functions can be realized as transfer functions of minimal passive selfadjoint discrete-time systems.We give various characterizations for the class \(\mathcal{RS}{\mathfrak(m)}\) and obtain an explicit form for the inner functions from the class \(\mathcal{RS}{\mathfrak(m)}\) as well as an inner dilation for any function from \(\mathcal{RS}{\mathfrak(m)}\). We also consider various transformations of the class \(\mathcal{RS}{\mathfrak(m)}\), construct realizations of their images, and find corresponding fixed points.

We study analytic and geometric properties of Stieltjes and inverse Stieltjes families defined on a separable Hilbert space and establish various minimal representations for them by means of compressed resolvents of various types of linear relations. Also attention is paid to some new peculiar properties of Stieltjes and inverse Stieltjes families, including an analog for the notion of inner functions which will be characterized in an explicit manner. In addition, families which admit different types of scale invariance properties are described. Two transformers that naturally appear in the Stieltjes and inverse Stieltjes classes are introduced and their fixed points are identified., 33 pages

The well-known results of M. G. Kreĭn concerning the description of selfadjoint contractive extensions of a hermitian contraction \(T_1\) and the characterization of all nonnegative selfadjoint extensions \({{\widetilde{A}} }\) of a nonnegative operator A via the inequalities \(A_K\le {{\widetilde{A}} } \le A_F\), where \(A_K\) and \(A_F\) are the Kreĭn–von Neumann extension and the Friedrichs extension of A, are generalized to the situation, where \({{\widetilde{A}} }\) is allowed to have a fixed number of negative eigenvalues. These generalizations are shown to be possible under a certain minimality condition on the negative index of the operators \(I-T_1^*T_1\) and A, respectively; these conditions are automatically satisfied if \(T_1\) is contractive or A is nonnegative, respectively. The approach developed in this paper starts by establishing first a generalization of an old result due to Yu. L. Shmul’yan on completions of nonnegative block operators. The extension of this fundamental result allows us to prove analogs of the above mentioned results of M. G. Kreĭn and, in addition, to solve some related lifting problems for J-contractive operators in Hilbert, Pontryagin and Kreĭn spaces in a simple manner. Also some new factorization results are derived, for instance, a generalization of the well-known Douglas factorization of Hilbert space operators. In the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non-contractive selfadjoint operators under Cayley transforms properly.

A linear relation, i.e., a multivalued operator $T$ from a Hilbert space ${\mathfrak H}$ to a Hilbert space ${\mathfrak K}$ has Lebesgue type decompositions $T=T_{1}+T_{2}$, where $T_{1}$ is a closable operator and $T_{2}$ is an operator or relation which is singular. There is one canonical decomposition, called the Lebesgue decomposition of $T$, whose closable part is characterized by its maximality among all closable parts in the sense of domination. All Lebesgue type decompositions are parametrized, which also leads to necessary and sufficient conditions for the uniqueness of such decompositions. Similar results are given for weak Lebesgue type decompositions, where $T_1$ is just an operator without being necessarily closable. Moreover, closability is characterized in different useful ways. In the special case of range space relations the above decompositions may be applied when dealing with pairs of (nonnegative) bounded operators and nonnegative forms as well as in the classical framework of positive measures., Comment: 33 pages

The theory of closed sesquilinear forms in the non-semi-bounded situation exhibits some new features, as opposed to the semi-bounded situation. In particular, there can be more than one closed form associated with the generalized Friedrichs extension SF of a non-semi-bounded symmetric operator S (if SF exists). However, there is one unique form [·, ·] satisfying Kato's second representation theorem and, in particular, dom = dom ∣SF∣1/2. In the present paper, another closed form [·, ·], also uniquely associated with SF, is constructed. The relation between these two forms is analysed and it is shown that these two non-semi-bounded forms can indeed differ from each other. Some general criteria for their equality are established. The results induce solutions to some open problems concerning generalized Friedrichs extensions and complete some earlier results about them in the literature. The study is connected to the spectral functions of definitizable operators in Kreĭn spaces.

Let H 1 H_1 and H 2 H_2 be selfadjoint operators or relations (multivalued operators) acting on a separable Hilbert space and assume that the inequality H 1 ≤ H 2 H_1 \leq H_2 holds. Then the validity of the inequalities − H 1 − 1 ≤ − H 2 − 1 -H_1^{-1} \le -H_2^{-1} and H 2 − 1 ≤ H 1 − 1 H_2^{-1} \le H_1^{-1} is characterized in terms of the inertia of H 1 H_1 and H 2 H_2 . Such results are known for matrices and boundedly invertible operators. In the present paper those results are extended to selfadjoint, in general unbounded, not necessarily boundedly invertible, operators and, more generally, for selfadjoint relations in separable Hilbert spaces.

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.

Let A be a densely defined simple symmetric operator in $${\mathfrak{H}}$$ , let $${\Pi=\{\mathcal{H},\Gamma_0, \Gamma_1}\}$$ be a boundary triplet for A * and let M(·) be the corresponding Weyl function. It is known that the Weyl function M(·) determines the boundary triplet Π, in particular, the pair {A, A 0}, uniquely up to the unitary similarity. Here $${A_0 := A^* \upharpoonright \text{ker}\, \Gamma_0 ( = A^*_0)}$$ . At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to the weak similarity. We consider a symmetric dual pair {A, A} with symmetric $${A \subset A^*}$$ and a special boundary triplet $${\widetilde{\Pi}}$$ for{A, A} such that the corresponding Weyl function is $${\widetilde{M}(z) = K^*(B-M(z))^{-1} K}$$ , where B is a non-self-adjoint bounded operator in $${\mathcal{H}}$$ . We are interested in the problem whether the result on the unitary similarity remains valid for $${\widetilde{M}(\cdot)}$$ in place of M(·). We indicate some sufficient conditions in terms of the operators A 0 and $${A_B= A^* \upharpoonright \text{ker}\, (\Gamma_1-B \Gamma_0)}$$ , which guaranty an affirmative answer to this problem. Applying the abstract results to the minimal symmetric 2nth order ordinary differential operator A in $${L^2(\mathbb{R}_+)}$$ , we show that $${\widetilde{M}(\cdot)}$$ defined in $${\Omega_+ \subset \mathbb{C}_+}$$ determines the Dirichlet and Neumann realizations uniquely up to the unitary equivalence. At the same time similar result for realizations of Dirac operator fails. We obtain also some negative abstract results demonstrating that in general the Weyl function $${\widetilde{M}(\cdot)}$$ does not determine A B even up to the similarity.

The paper is devoted to the development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. We mainly focus on the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results involve boundary value techniques and reproducing kernel space methods, and they include an explicit functional model for the class of stable C-symmetries. Some of the results are specialized further by studying the case where S has defect numbers 〈2,2〉 in detail.

Truncated 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.

The basic objects in this paper are monotonically nondecreasing n × n matrix functions D ( · ) defined on some open interval i = ( a , b ) of R and their limit values D ( a ) and D ( b ) at the endpoints a and b which are, in general, selfadjoint relations in C n . Certain space decompositions induced by the matrix function D ( · ) are made explicit by means of the limit values D ( a ) and D ( b ) . They are a consequence of operator inequalities involving these limit values and the notion of strictness (or definiteness) of monotonically nondecreasing matrix functions. This treatment provides a geometric approach to the square-integrability of solutions of definite canonical systems of differential equations.

Suppose that A is a symmetric densely defined operator in a Hilbert space H with equal deficiency indices. In the last 30 years, the approach based on the notions of boundary triplet of an operatorA∗ and of the corresponding Weyl function has become quite popular in extension theory (see [1]–[3]). It is well known (see [2]) that, in the case a simple operator A, the Weyl function M( · ) of the boundary triplet Π = {H ,Γ0,Γ1} uniquely determines this triplet up to unitary equivalence. In particular,M( · ) defines the pair {A,A0}, in which A0 := A∗ ker Γ0 = A0 is uniquely defined up to unitary similarity. On the other hand, if ΠD = {H0 ⊕H1,Γ,Γ } is the boundary triplet of the dual pair {A,A } of operators, then the corresponding Weyl function defines the triplet ΠD only up to weak similarity (see [4]). The conditions on the pair {A,A }, under which the Weyl function defines the triplet ΠD up to similarity were obtained in the recent papers [5]–[7].

Boundary value problems for singular canonical systems of differential equations of the form Jf'(t) - H(t)f(t) = lambda Delta(t)f(t), t is an element of i, lambda is an element of C, are studied in the associated Hilbert space L(Delta)(2)(i). With the help of a monotonicity principle for matrix functions their square-integrable solutions are specified. This yields a direct treatment of defect numbers of the minimal relation and simultaneously makes it possible to assign certain boundary values to the elements of the maximal relation induced by the system of differential equations in L(Delta)(2)(i). The investigation of boundary value problems for these systems and their spectral theory can be carried out by means of abstract boundary triplet techniques. This paper makes explicit the construction and the properties of boundary triplets and Weyl functions for singular canonical systems. Furthermore, the Weyl functions are shown to have a property similar to that of the classical Titchmarsh-Weyl coefficients for singular Sturm-Liouville operators: they single out the square-integrable solutions of the homogeneous systems of canonical differential equations. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Boundary relations for a symmetric relation in a Pontryagin space are studied and the corresponding Weyl families are characterized. In particular, it is shown that every generalized Nevanlinna family can be realized as the Weyl family of a boundary relation in a Pontryagin space. Mathematics subject classification (2010): 46C20; 46E22; 47B50, Secondary: 47A48; 47B32.

Let −DpD be a differential operator on the compact interval [−b, b] whose leading coefficient is positive on (0, b] and negative on [−b, 0), with fixed, separated, self-adjoint boundary conditions at b and −b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal sesquilinear form. Moreover, among all closed forms associated with the self-adjoint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Kreĭn–Feller operators with a single concentrated mass.

A nonnegative form t on a complex linear space is decomposed with respect to another nonnegative form tu: it has a Lebesgue decomposition into an almost dominated form and a singular form. The part which is almost dominated is the largest form majorized by t which is almost dominated by w. The construction of the Lebesgue decomposition only involves notions from the complex linear space. An important ingredient in the construction is the new concept of the parallel sum of forms. By means of Hilbert space techniques the almost dominated and the singular parts are identified with the regular and a singular parts of the form. This decomposition addresses a problem posed by B. Simon. The Lebesgue decomposition of a pair of finite measures corresponds to the present decomposition of the forms which are induced by the measures. T. Ando's decomposition of a nonnegative bounded linear operator in a Hilbert space with respect to another nonnegative bounded linear operator is a consequence. It is shown that the decomposition of positive definite kernels involving families of forms also belongs to the present context. The Lebesgue decomposition is an example of a Lebesgue type decomposition, i.e., any decomposition into an almost dominated and a singular part. There is a necessary and sufficient condition for a Lebesgue type decomposition to be unique. This condition is inspired by the work of Ando concerning uniqueness questions. (C) 2009 Elsevier Inc. All rights reserved.

A completeness theorem is proved involving a system of integro-differential equations with some λ-depending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established.

Passive systems \(\tau = \{T,{\mathfrak{M}},{\mathfrak{N}},{\mathfrak{H}}\}\) with \({\mathfrak{M}}\) and \({\mathfrak{N}}\) as an input and output space and \({\mathfrak{H}}\) as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system \(\tau\) with \({\mathfrak{M}} = {\mathfrak{N}}\) is said to be quasi-selfadjoint if ran \((T - T^*) \subset {\mathfrak{N}}\). The subclass \({\bf S}^{qs}({\mathfrak{N}})\) of the Schur class \({\bf S}({\mathfrak{N}})\) is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass \({\bf S}^{qs}({\mathfrak{N}})\) is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass \({\bf S}^{qs}({\mathfrak{N}})\) and the Q-function of T is given.

Recently a new notion, the so-called boundary relation, has been introduced involving an analytic object, the so-called Weyl family. Weyl families and boundary relations establish a link between the class of Nevanlinna families and unitary relations acting from one Krein in space, a basic (state) space, to another Krein in space, a parameter space where the Nevanlinna family or Weyl family is acting. Nevanlinna families are a natural generalization of the class of operator-valued Nevanlinna functions and they are closely connected with the class of operator-valued Schur functions. This paper establishes the connection between boundary relations and their Weyl families on the one hand, and unitary colligations and their transfer functions on the other hand. From this connection there are various advances which will benefit the investigations on both sides, including operator theoretic as well as analytic aspects. As one of the main consequences a functional model for Nevanlinna families is obtained from a variant of the functional model of L. de Branges and J. Rovnyak for Schur functions. Here the model space is a reproducing kernel Hilbert space in which multiplication by the independent variable defines a closed simple symmetric operator. This operator gives rise to a boundary relation such that the given Nevanlinna family is realized as the corresponding Weyl family.

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

Passive linear systems τ = $${\{A, B, C, D; \mathfrak{H}, \mathfrak{M}, \mathfrak{N}\}}$$ have their transfer function $${\Theta_{\tau}(\lambda) = D + \lambda C{(I - \lambda A)^{-1}}B}$$ in the Schur class S $${(\mathfrak{M}, \mathfrak{N})}$$ . Using a parametrization of contractive block operators the transfer function $${\Theta_{\tau}(\lambda)}$$ is connected to the Sz.-Nagy–Foias characteristic function $${\Phi_{A}(\lambda)}$$ of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions $${\Theta_{\tau}(\lambda)}$$ and $${\Phi_{A}(\lambda)}$$ . The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained.

The sum of two unbounded nonnegative selfadjoint operators is a nonnegative operator which is not necessarily densely defined. In general its selfadjoint extensions exist in the sense of linear relations (multivalued operators). One of its nonnegative selfadjoint extensions is constructed via the form sum associated with A and B. Its relations to the Friedrichs and Krein--von Neumann extensions of A+Bare investigated. For this purpose, the one-to-one correspondence between densely defined closed semibounded forms and semibounded selfadjoint operators is extended to the case of nondensely defined semibounded forms by replacing semibounded selfadjoint operators by semibounded selfadjoint relations. In particular, the inequality between two closed nonnegative forms is shown to be equivalent to a similar inequality between the corresponding nonnegative selfadjoint relations.

is complete in the space L[0, 1] for any complex-valued potential q ∈ L[0, 1] and any h0 , h1 ∈ C . In the same paper [1], the completeness of the system of eigenfunctions and associated functions is established in the case of an arbitrary (rather than just separable) nondegenerate boundary conditions. The completeness of the system of eigenfunctions and associated functions of the differential equation

In this paper operator-valued Q-functions of Krein-Ovcharenko type are introduced. Such functions arise from the extension theory of Hermitian contractive operators A in a Hilbert space h. The definition is related to the investigations of M.G. Krein and I.E. Ovcharenko of the so-called Q(mu) and Q(M)-functions. It turns out that their characterizations of such functions hold true only in the matrix valued case. The present paper extends the corresponding properties for wider classes of selfadjoint contractive extensions of A. For this purpose some peculiar but fundamental properties on the behaviour of operator ranges of positive operators will be used. Also proper characterizations for Q(mu)- and Q(M)-functions in the general operator-valued case are given. Shorted operators and parallel sums of positive operators will be needed to give a geometric understanding of the function-theoretic properties of the corresponding Q-functions.

Let -Domega((.), z)D + q be a differential operator in L-2(0, infinity) whose leading coefficient contains the eigenvalue parameter z. For the case that omega((.), z) has the particular formomega(t, z) = p(t) + c(t)(2)/(z - r (t)), z is an element of C \ R,and the coefficient functions satisfy certain local integrability conditions, it is shown that there is an analog for the usual limit-point/limit-circle classification. In the limit-point case mild sufficient conditions are given so that all but one of the Titchmarsh-Weyl coefficients belong to the so-called Kac subclass of Nevanlinna functions. An interpretation of the Titchmarsh-Weyl coefficients is given also in terms of an associated system of differential equations where the eigenvalue parameter appears linearly. (C) 2004 Elsevier Inc. All rights reserved.

In M. G. Krein's extension theory of nonnegative operators a complete description is given of all nonnegative selfadjoint extensions of a densely defined nonnegative operator. This theory, the refinements to the theory due to T. Ando and K. Nishio, and its extension to the case of nondensely defined nonnegative operators is being presented in a unified way, building on the completion of nonnegative operator blocks. The completion of nonnegative operator blocks gives rise to a description of all selfadjoint contractive extensions of a symmetric (nonselfadjoint) contraction. This in turn is equivalent to a description of all nonnegative selfadjoint relation (multivalued operator) extensions of a nonnegative relation. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.