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In the first part of this note we give a rather short proof of a generalization of Stenger’s lemma about the compression $$A_0$$ to $${{\mathfrak {H}}}_0$$ of a self-adjoint operator A in some Hilbert space $${{\mathfrak {H}}}={{\mathfrak {H}}}_0\oplus {{\mathfrak {H}}}_1$$ . In this situation, $$S:=A\cap A_0$$ is a symmetry in $${{\mathfrak {H}}}_0$$ with the canonical self-adjoint extension $$A_0$$ and the self-adjoint extension A with exit into $${{\mathfrak {H}}}$$ . In the second part we consider relations between the resolvents of A and $$A_0$$ like M.G. Krein’s resolvent formula, and corresponding operator models.

Let S be a symmetric relation with finite and equal defect numbers in the Hilbert space \(\mathfrak H\). If \(\widetilde A\) is a self-adjoint extension of S in some larger Hilbert space \(\widetilde {\mathfrak H}\), the compression of \(\widetilde A\) to \({\mathfrak H}\) is a symmetric extension of S. We study this compression in dependence of the parameter \({\mathcal T}\), which parametrizes the extensions \(\widetilde A\) according to M.G. Krein’s resolvent formula. By means of a fractional transformation, analogous results are proved for the Straus extensions of S at a real point.

We establish new analytic results for a general class of rational spectral problems. They arise e.g. in modelling photonic crystals whose capability to control the flow of light depends on specific ...

In this paper we establish conditions for the existence of maximal J -semi-definite invariant subspaces of unbounded J -selfadjoint operators. Our results allow for operators where all entries of the formal matrix representation induced by the indefinite metric are unbounded and they do not require any definiteness or J -dissipativity assumptions. As a consequence of the existence of invariant subspaces, we obtain an unexpected result on the accumulation of non-real eigenvalues at the real axis which is of independent interest. An application to some dissipative two-channel Hamiltonians illustrates this new phenomenon.

In this paper a two-step reduction method for spectral problems on a star graph with n+1 edges e_{0}, e_{1}, \ldots , e_{n} and a self-adjoint matching condition at the central vertex v is established. The first step is a reduction to the problem on the single edge e_0 but with an energy depending boundary condition at v. In the second step, by means of an abstract inverse result for Q-functions, a reduction to a problem on a path graph with two edges e_0, \widetilde{e}_1 joined by continuity and Kirchhoff conditions is given. All results are proved for symmetric linear relations in an orthogonal sum of Hilbert spaces. This ensures wide applicability to various different realizations, in particular, to canonical systems and Krein strings which include, as special cases, Dirac systems and Stieltjes strings. Employing two other key inverse results by de Branges and Krein, we answer e.g. the following question: If all differential operators are of one type, when can the reduced system be chosen to consist of two differential operators of the same type?

Let S be a symmetric operator with finite and equal defect numbers in the Hilbert space $${{\mathfrak {H}}}$$ . We study the compressions $$P_{{\mathfrak {H}}}\widetilde{A}\big |_{{\mathfrak {H}}}$$ of the self-adjoint extensions $$\widetilde{A}$$ of S in some Hilbert space $$\widetilde{{\mathfrak {H}}}\supset {{\mathfrak {H}}}$$ . These compressions are symmetric extensions of S in $${{\mathfrak {H}}}$$ . We characterize properties of these compressions through the corresponding parameter of $$\widetilde{A}$$ in M.G. Krein’s resolvent formula. If $$\dim \, (\widetilde{{\mathfrak {H}}}\ominus {{\mathfrak {H}}})$$ is finite, according to Stenger’s lemma the compression of $$\widetilde{A}$$ is self-adjoint. In this case we express the corresponding parameter for the compression of $$\widetilde{A}$$ in Krein’s formula through the parameter of the self-adjoint extension $$\widetilde{A}$$ .

In this paper the spectral properties of the abstract Klein–Gordon equation are studied. The main tool is an indefinite inner product known as the charge inner product. Under certain assumptions on the potential V, two operators are associated with the Klein–Gordon equation and studied in Krein spaces generated by the charge inner product. It is shown that the operators are self-adjoint and definitizable in these Krein spaces. As a consequence, they possess spectral functions with singularities, their essential spectra are real with a gap around 0 and their non-real spectra consist of finitely many eigenvalues of finite algebraic multiplicity which are symmetric to the real axis. One of these operators generates a strongly continuous group of unitary operators in the Krein space; the other one gives rise to two bounded semi-groups. Finally, the results are applied to the Klein–Gordon equation in ℝn.

We study the intimate relation of the continuation problem for hermitian positive definite functions with direct and inverse spectral problems for canonical systems and strings and with the extrapolation problem for second order stochastic processes.

Let S be a symmetric operator with finite and equal defect numbers d in the Hilbert space \(\mathfrak{H}\), and with a boundary triplet \((\mathbb{C}^d, \Gamma_1, \Gamma_2)\). Following the method of E.A. Coddington, we describe all self-adjoint extensions \(\tilde{A}\) of S in a Hilbert space \(\tilde{\mathfrak{H}}\;=\;\mathfrak{H}\oplus \mathfrak{H}_1\) where \(\mathrm{dim}\;\mathfrak{H}_1\

It is shown that transfer functions, which play a crucial role in M.G. Krein's study of inverse spectral problems, are a proper tool to formulate local spectral uniqueness conditions., 23 pages, 3 figures

We consider a Bessel-type differential expression on [0, a], a > 1, with the singularity at the inner point x = 1, see (1.2) below. This singularity is in the limit point case from both sides. Therefore in a Hilbert space treatment in L2(0, a), e.g. for Dirichlet boundary conditions at x = 0 and x = a, a unique self-adjoint operator is associated with this differential expression. However, in papers by J. F. van Diejen and A. Tip, Yu. Shondin, A. Dijksma, P. Kurasov and others, in more general situations, self-adjoint operators in some Pontryagin space were connected with this kind of singular equations; for (1.2) this connection appeared also in the study of a continuation problem for a hermitian function by H. Langer, M. Langer and Z. Sasv´ari. In the present paper we give an explicit construction of this Pontryagin space for the Besseltype equation (1.2) and a description of the self-adjoint operators which can be associated with it.

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. (c) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

We solve a boundary interpolation problem at a real point for generalized Nevanlinna functions, and use the result to prove uniqueness theorems for generalized Nevanlinna functions. (C) 2010 WILEY-VCH Verlag GmbH & Co KGaA. Weinheim

The Titchmarsh–Weyl function, which was introduced in Fulton (Math Nachr 281(10):1418–1475, 2008) for the Sturm-Liouville equation with a hydrogen-like potential on (0, ∞), is shown to belong to a generalized Nevanlinna class $${\bf N_\kappa}$$ . As a consequence, also in the case of two singular endpoints for the Fourier transformation defined by means of Frobenius solutions there exists a scalar spectral function. This spectral function is given explicitly for potentials of the form $${\dfrac{q_0}{x^2}+\dfrac{q_1}{x},\,-\dfrac 14\le q_0 < \infty}$$ .

We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph $\mathcal G$ with a finite number $n$ of edges. This operator is defined by a Dirac-Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of $\mathcal G$. Special attention is paid to Robin matching conditions with parameter $\tau \in\mathbb R\cup\{\infty\}$. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein's resolvent formula, introduce corresponding Weyl-Titchmarsh functions, study the multiplicities, dependence on $\tau$, and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for $R\to \infty$, the difference of the number of eigenvalues in the intervals $[0,R)$ and $[-R,0)$ deviates from some integer $\kappa_0$, which we call dislocation index, at most by $n+2$., Comment: Accepted for publication in IEOT

We establish sufficient conditions for the so-called Virozub–Matsaev condition for twice continuously differentiable self-adjoint operator functions. This condition is closely related to the existence of a local spectral function and to the notion of positive type spectrum. Applications to self-adjoint operators in Krein spaces and to quadratic operator polynomials are given.

We introduce a Schur transformation for generalized Nevanlinna functions and show that it can be used in obtaining the unique minimal factorization of a class of rational J(l)-unitary 2 x 2 matrix functions into elementary factors from the same class. (c) 2006 Elsevier Inc. All rights reserved.

For a self-adjoint operator in a Krein space we construct an interval [ν, μ] outside of which the operator has only a spectrum of definite type and possesses a local spectral function. As a consequence, a spectral subspace corresponding to an interval outside [ν, μ] admits an angular operator representation. We describe a defect subspace of the domain of the angular operator in terms of the Schur complement, and we derive variational principles for the discrete eigenvalues in such intervals of definite type.

We prove oscillation results for the real eigenvalues of Sturm–Liouville problems with an indefinite weight function. An essential role is played by the signature of an eigenvalue, which is shown to be related to the signs of the corresponding leading coefficients of the Titchmarsh–Weyl m-function and of the Prüfer angle at this eigenvalue.

In this note we apply Krein space methods to PT-symmetric problems to obtain conditions for the spectrum to be real and estimates of the number of non-real spectral points.

We prove that a 2×2 matrix polynomial which is J-unitary on the real line can be written as a product of normalized elementary J-unitary factors and a J-unitary constant. In the second part we give an algorithm for this factorization using an analog of the Schur transformation.

In this paper we relate the operators in the operator representations of a generalized Nevanlinna function N(z) and of the function −N(z)−1 under the assumption that z=∞ is the only (generalized) pole of nonpositive type. The results are applied to the Q-function for S and H and the Q-function for S and H∞, where H is a self-adjoint operator in a Pontryagin space with a cyclic element w, H∞ is the self-adjoint relation obtained from H and w via a rank one perturbation at infinite coupling, and S is the symmetric operator given by S=H∩H∞.

The main result is that forJ = ((1)(0) (0)(-1))every J-unitary 2 x 2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2 x 2-matrix polynomials which are either of degree 1 or 2k. This is shown by means of the generalized Schur transformation introduced in [Ann. Inst. Fourier 8 (1958) 211; Ann. Acad. Sci. Fenn. Ser. A I 250 (9) (1958) 1-7] and studied in [Pisot and Salem Numbers, Birkhauser Verlag, Basel, 1992; Philips J. Res. 41 (1) (1986) 1-54], and also in the first two parts [Operator Theory: Adv. Appl. 129, Birkhauser Verlag, Basel, 2000, p. 1; Monatshefte fur Mathematik, in press] of this series. The essential tool in this paper are the reproducing kernel Pontryagin spaces associated with generalized Schur functions. (C) 2003 Elsevier Science Inc. All fights reserved.

In this paper self-adjoint 2×2 block operator matrices A in a Hilbert space H1⊕H2 are considered. For an interval Δ which does not intersect the spectrum of at least one of the diagonal entries of A, we prove angular operator representations for the corresponding spectral subspace LΔ(A) of A and we study the supporting subspace in this angular operator representation of LΔ(A), which is the orthogonal projection of LΔ(A) to the corresponding component H1 or H2. Our main result is a description of a special direct complement of this supporting subspace in its component in terms of spectral subspaces of the values of the corresponding Schur complement of A in the endpoints of Δ.

In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1-36) it was shown that for a generalized Schur function s(z), which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal s(z)(-1) of the characteristic function s(z) and general factorization results for characteristic functions.

In this paper variational principles for block operator matrices are established which are based on functionals associated with the quadratic numerical range. These principles allow to characterize, e.g., eigenvalues in gaps of the essential spectrum and to derive two-sided eigenvalue estimates in terms of the spectral characteristics of the entries of the block operator matrix. The results are applied to a second order partial differential equation depending on the spectral parameter nonlinearly.

Loewner's theorem on boundary interpolation of N κ functions is proved under rather general conditions. In particular, the hypothesis of Alpay and Rovnyak (1999) that the function f, which is to be extended to an N κ function, is defined and continuously differentiable on a nonempty open subset of the real line, is replaced by the hypothesis that the set on which f is defined contains an accumulation point at which f satisfies some kind of differentiability condition. The proof of the theorem in this note uses the representation of N κ functions in terms of selfadjoint relations in Pontryagin spaces and the extension theory of symmetric relations in Pontryagin spaces.

In this paper we consider the Sturm–Liouville operator d2/dx2 − 1/x on the interval [a, b], a < 0 < b, with Dirichlet boundary conditions at a and b, for which x = 0 is a singular point. In the two components L2(a, 0) and L2(0, b) of the space L2(a, b) = L2(a, 0) ⊕ L2(0, b) we define minimal symmetric operators and describe all the maximal dissipative and self-adjoint extensions of their orthogonal sum in L2(a, b) by interface conditions at x = 0. We prove that the maximal dissipative extensions whose domain contains only continuous functions f are characterized by the interface condition limx→0+(f′(x)−f′(−x)) = γf(0) with γ∈C+∪R or by the Dirichlet condition f(0+) = f(0−) = 0. We also show that the corresponding operators can be obtained by norm resolvent approximation from operators where the potential 1/x is replaced by a continuous function, and that their eigen and associated functions can be chosen to form a Bari basis in L2(a, b).

To describe the dynamics of a single peak of the Rosensweig instability a model is proposed which approximates the peak by a half-ellipsoid atop a layer of magnetic fluid. The resulting nonlinear equation for the height of the peak leads to the correct subcritical character of the bifurcation for static induction. For a time-dependent induction the effects of inertia and damping are incorporated. The results of the model show qualitative agreement with the experimental findings, as in the appearance of period doubling, trebling, and higher multiples of the driving period. Furthermore a quantitative agreement is also found for the parameter ranges of frequency and induction in which these phenomena occur., Comment: 21 pages, 9 figures, using elsart, submitted to Physica D; revised version with 2 figures and references added

Double variational principles are established for eigenvalues of a (norm) continuous self-adjoint operator valued functionL defined on a real interval [α, β[.L(λ) is not required to be definite for any λ. Applications are made to linear, quadratic and rational functionsL.

LetQ∈N k. It is shown that if α is a nonreal pole or a real generalized pole of nonpositive type and β is a nonreal zero or a real generalized zero of nonpositive type of the functionQ then the function $$Q_1 (z): = \frac{{(z - \alpha )(z - \bar \alpha )}}{{(z - \beta )(z - \bar \beta )}}Q(z)$$ belongs to the classN k−1.

We give an example of a positive operator B in a Krein space with the following properties: the nonzero spectrum of B consists of isolated simple eigenvalues, the norms of the orthogonal spectral projections in the Krein space onto the eigenspaces of B are uniformly bounded and the point ∞ is a singular critical point of B. An operator A in the Krein space (K, [ · , · ]) is said to be positive if [Ax, x] > 0 for all nonzero x in the domain of A. A bounded positive operator A in the Krein space (K, [ · , · ]) has a projection valued spectral function E with 0 being its only possible critical point (see [1, Theorem IV.1.5] or [5, Section II.3.]). Recall that, by [5, Proposition 5.6], the condition ‖E((−∞, α])‖ ≤ C− 0 (2) is equivalent to the existence of the limit limβ↓0E([β,+∞)) in the strong operator topology. Since 0 is not an eigenvalue of a positive operator A, [5, Proposition 3.2] implies that (1) and (2) are equivalent. Also, if 0 is a critical point, it is said to be regular if one of the conditions (1) or (2) is fulfilled. If the critical point 0 is not regular, it is called singular. In the sequel the operator A considered will have a discrete spectrum outside 0. Examples of bounded positive operators inK having 0 as a singular critical point can be constructed as follows (see also the examples in [2, Section 1], [3], [4]). Consider a sequence of two-dimensional Krein spaces Kn = C with fundamental symmetry Jn = ( 1 0 0 −1 ) and positive operators An in Kn; denote by λn (λn , respectively) its positive (negative, respectively) eigenvalues and by P n (P − n , respectively) the orthogonal (in Kn) projection onto the corresponding eigenspace. Received by the editors October 15, 1998. 2000 Mathematics Subject Classification. Primary 47B50, 46C50.

Let the functionQ be holomorphic in he upper half plane ℂ+ and such that ImQ(z ≥ 0 and ImzQ(z) ≥ 0 ifz e ℂ+. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf +λ fdm = 0; f′(0−) = 0. In this note we show that the set of functionsQ which are holomorphic in ℂ+ and such that the kernel $$\frac{{Q(z) - \overline {Q(\zeta )} }}{{z - \bar \zeta }}$$ hasκ negative squares of ℂ+ and ImzQ(z) ≥ 0 ifz e ℂ+ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf′ +λf dm +λ 2 fdD = 0 on [0,L),f′(0−) = 0. Hereκ is the number of pointsx whereD increases or 0 >m(x + 0) −m(x − 0) ≥ −∞; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.

LetA be a selfadjoint definitizable operator in a Krein space. It is shown that there exists a finite rank nonnegative perturbation ofA of arbitrarily small norm such that all the singular critical points ofA of finite index disappear.

In this paper, we study the Orr–Sommerfeld problem on a finite interval. It is shown that the eigenfunctions and associated functions form a Bari basis in a suitable Hilbert space if the unperturbed velocity profile u is sufficiently smooth. To this end, the Orr–Sommerfeld problem is considered as a bounded perturbation of a certain self-adjoint spectral problem.

Spectral properties of the sum of a linear span of projections and a compact nonnegative operator are considered. In particular, we prove partial completeness results for certain parts of the system of eigenfunctions. The main tool is a transformation the original spectral problem to that of a monic weakly hyperbolic pencil.

In this note a matrix partial differential operator is considered. It is shown that under certain conditions it defines a closed operator with nonempty resolvent set, and its essential spectrum is determined. In the symmetric case G.D. Raikov obtained earlier corresponding results (under slightly different assumptions) for the Friedrichs extension of the operator.

In this paper we continue the study of spectral properties of a selfadjoint analytic operator function A((z)) under the Virozub-Matsaev condition. As in [6], [7], main tools are the linearization and the factorization of A(z).We use an abstract definition of a so-called Hilbert space linearization and show its uniqueness, and we prove a generalization of the well-known factorization theorem from [10]. The main results concern properties of the compression AΔ((z)) of A((z)) to its spectral subspace, called spectral compression of A((z)). Close connections between the linearization, the inner linearization, and the local spectral function of A((z)) and of its spectral compression AΔ(z) are established

It is shown that M. Krein's method of directing functionals can be used to prove the existence of a scalar spectral measure for certain Sturm-Liouville equations with two singular endpoints. The essential assumption is the existence of a solution of the equation that is square integrable at one singular endpoint and depends analytically on the eigenvalue parameter., 8 pages

This note contains some supplements to our earlier notes [LN II], [LN III], where the Newton diagram was used in order to obtain in a straightforward way information about the perturbed eigenvalues of an analytic and analytically perturbed matrix function.