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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.

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.

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

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 existence of magnetohydrodynamic mean-field alpha^2-dynamos with spherically symmetric, isotropic helical turbulence function alpha is related to a non-self-adjoint spectral problem for a coupled system of two singular second order ordinary differential equations. We establish global estimates for the eigenvalues of this system in terms of the turbulence function alpha and its derivative alpha'. They allow us to formulate an anti-dynamo theorem and a non-oscillation theorem. The conditions of these theorems, which again involve alpha and alpha', must be violated in order to reach supercritical or oscillatory regimes., Comment: 35 pages, 4 figures, to be published in SIAM J. Math. Anal.

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 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.

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.

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.

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

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.

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

A Nevanlinna function is a function which is analytic in the open upper half-plane and has a non-negative imaginary paxt there. In this paper we study a fractional linear transformation for a Nevanlinna function n with a suitable asymptotic expansion at ∞, that is an analogue of the Schur transformation for contractive analytic functions in the unit disk. Applying the transformation p times we find a Nevanlinna function n p which is a fractional linear transformation of the given function n. The main results concern the effect of this transformation to the realizations of n and n p by which we mean their representations through resolvents of self-adjoint operators in Hilbert space. Our tools are block operator matrix representations, u-resolvent matrices, and reproducing kernel Hilbert spaces.

We review our recent work on the Schur transformation for scalar generalized Schur and Nevanlinna functions. The Schur transformation is defined for these classes of functions in several situations, and it is used to solve corresponding basic interpolation problems and problems of factorization of rational J-unitary matrix functions into elementary factors. A key role is played by the theory of reproducing kernel Pontryagin spaces and linear relations in these spaces.

We define and solve a boundary interpolation problem for generalized Schur functions s(z) on the open unit disk \( \mathbb{D}\) which have preassigned asymptotics when z from \( \mathbb{D}\) tends nontangentially to a boundary point z1 ∈ \( \mathbb{T}\). The solutions are characterized via a fractional linear parametrization formula. We also prove that a rational J-unitary 2 × 2-matrix function whose only pole is at z1 has a unique minimal factorization into elementary factors and we classify these factors. The parametrization formula is then used in an algorithm for obtaining this factorization. In the proofs we use reproducing kernel space methods.

In this paper we investigate an abstract Klein–Gordon equation by means of indefinite inner product methods. We show that, under certain assumptions on the potential which are more general than in previous works, the corresponding linear operator A is self-adjoint in the Pontryagin space $$\mathcal{K}$$ induced by the so-called energy inner product. The operator A possesses a spectral function with critical points, the essential spectrum of A is real with a gap around 0, and the non-real spectrum consists of at most finitely many pairs of complex conjugate eigenvalues of finite algebraic multiplicity; the number of these pairs is related to the ‘size’ of the potential. Moreover, A generates a group of bounded unitary operators in the Pontryagin space $$\mathcal{K}$$ . Finally, the conditions on the potential required in the paper are illustrated for the Klein–Gordon equation in $$\mathbb{R}^n$$ ; they include potentials consisting of a Coulomb part and an L p -part with n ≤ p < ∞.

For a special class 2 × 2-matrix functions Ω operator representations of Ω(z) and Ω^(z) := - Ω(z) -1 by means of self-adjoint linear relations A and A^ in a Krein space K are given. Since A^ is a 2-dimensional perturbation of A, results of Langer, Markus and Matsaev imply that "singularities of positive type" of Ω remain singularities of positive type of Ω^ with the possible exception of isolated points which have a "finite negative index".

Nephroprotection by long-term ACE inhibition with ramipril in spontaneously hypertensive stroke prone rats. Background The effect of life-long treatment with the ACE inhibitor ramipril on hypertension-induced histological changes in the kidney was tested in stroke-prone spontaneously hypertensive rats (SHR-SP). Methods One-month-old pre-hypertensive SHR-SP were randomized into three groups of 45 animals each, and exposed via drinking water for their lifetime to a dose of: 1mg · kg -1 · d -1 ramipril (antihypertensive dose, HRA); 10 μg · kg -1 · d -1 slight dose of ramipril (non-antihypertensive dose, LRA); or placebo. Histological and biochemical assessments were conducted after 15months in ten rats each, when about 80% of the placebo group had died. Results Kidneys from placebo treated SHR-SP showed pronounced arterial wall hypertrophy and sclerosis, arterial fibrinoid necrosis, glomerulopathy and tubular interstitial injury that were, in concert with normalized blood pressure, completely prevented by HRA treatment. LRA treatment did not affect any blood pressure increase, and also attenuated the development of arterial wall hypertrophy, sclerosis and arterial fibrinoid necrosis, though to a minor extent only, but did not change glomerular and tubulointerstitial degeneration. These effects of ramipril were associated with a dose-dependent inhibition of plasma and renal tissue ACE activities as well as lower serum concentrations of creatinine, but there were no changes in serum potassium. Conclusions Life-long HRA-induced ACE inhibition protects against hypertension-induced renal damages in SHR-SP. This is associated with a doubeling of the lifespan in these animals[1].

A classical theorem of Cauchy states that the eigenvalues of a principal submatrix A 0 of a Hermitian matrix A interlace the eigenvalues of A . We consider the case of a matrix A which is Hermitian with respect to an indefinite inner product.

In this paper a new concept for 2×2 -block operator matrices – the quadratic numerical range – is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.

For a self-adjoint analytic operator function A ( λ ) , which satisfies on some interval Δ of the real axis the Virozub–Matsaev condition, a local spectral function Q on Δ, the values of which are non-negative operators, is introduced and studied. In the particular case that A ( λ ) = λ I − A with a self-adjoint operator A, it coincides with the orthogonal spectral function of A. An essential tool is a linearization of A ( λ ) by means of a self-adjoint operator in some Krein space and the local spectral function of this linearization. The main results of the paper concern properties of the range of Q ( Δ ) and the description of a natural complement of this range.

Nonnegative Hermitian solutions of various types of continuous and discrete algebraic Riccati equations are studied. The Hamiltonian is considered with respect to two different indefinite scalar products. For the set of nonnegative solutions the order structure and the topology of the set and the stability of solutions is treated. For general Hermitian solutions a method to compute the inertia is given. Although most attention is payed to the classical types arising from LQ optimal control theory, the case where the quadratic term has an indefinite coefficient is studied as well.

On considere les proprietes spectrales du probleme differentiel: l(f)=(−1) n (p 0 f (n) ) (n) +(−1) n−1 (p 1 f (n−1) ) (n−1) +…+p n f=λrf sur un intervalle fini ou infini (a,b) avec des coefficients reels localement sommables 1/p 0 ,p 1 ,…,p n ,r sous l'hypothese que p 0 >0 et que la fonction poids r change de signe sur (a,b)

Let A be a uniformly elliptic second order linear operator on a smooth bounded domain Ω⊂ R n. We study the eigenvalue problem Au=λu subject to boundary conditions B0u=λB1u on ∂Ω, where Bj are linear boundary operators. The problem is recast in the form A u=λu in a Hilbert or Krein space, and results are given on the location and type of the spectrum, full- and half-range completeness, and regularity of critical points.

In this paper we establish criteria for the existence and uniqueness of contractive solutions K of the Riccati equation KBK + KA − DK − C =0 under the assumption that the spectra of A and D are disjoint.

S3226, a novel NHE3 inhibitor, attenuates ischemia-induced acute renal failure in rats. Background Acute renal failure (ARF) remains a major problem in clinical nephrology characterized by sudden loss of the kidney function due to ischemia, trauma, and/or nephrotoxic drugs. The current therapy of ARF is symptomatic with mortality rates exceeding 50%. The aim of this study was to investigate the effects of an intravenous infusion of S3226 (3-[2-(3-guanidino-2-methyl-3-oxopropenyl)-5-methyl-phenyl]-N-isopropylidene-2-methyl-acrylamide dihydrochloride), a selective Na + /H + exchange subtype 3 (NHE3) blocker, in ischemia-induced ARF in rats. In a second series of experiments cytosolic pH (pHi) changes in the kidney during ARF were continuously measured by means of nuclear magnetic resonance spectroscopy (MRS). Methods ARF was induced by bilateral occlusion of renal arteries for 40 minutes in three groups of anaesthetized Wistar rats. Control rats ( N = 12) were infused with saline (6.25 mL/kg over 30 min) before occlusion and the compound groups (each N = 12) were infused with S3226 at a dose of 20 mg/kg over 30 minutes either before initiation of ischemia or immediately after release of clamps. Plasma creatinine (P Cr ), creatinine clearance (C Cr ), urine volume, sodium, and potassium excretion were determined up to seven days after release of clamps. In the second series of experiments in anaesthetized rats the left kidney was exposed by flank incision and fixed in a non-magnetic device. An inflatable cuff was positioned around the pedicle to induce ischemia without removing animals from the magnet. A double-tuned 1 H- 31 P home-built surface coil was placed above the exposed kidney for the detection of pHi. Results At day 1 after ischemia C Cr in the control group was significantly lower as compared to S3226-treated animals (control 0.30 ± 0.05 vs. before 0.90 ± 0.26 and reperfusion 0.83 ± 0.15 mL/min/kg, respectively). P Cr increased from 18 ± 0.1 μmol/L before occlusion to 245 ± 7 μmol/L in the control. The increase in P Cr was significantly lower in the S3226 treated groups on days 1, 2, and 3 post-infusion. Fractional sodium excretion decreased significantly from 8.17% in the control to 1.42% and 1.88% in the treated groups. Renal pHi was significantly decreased by 0.15 units versus control during reperfusion. Histological examination of the kidneys on day 7 revealed pronounced reduction of tubular necrosis, dilatation, protein casts and cellular infiltration. Conclusions These results demonstrate that an intravenous administration of S3226 acutely improves GFR and kidney function and structure in both treated groups. In addition, in a separate set of studies S3226 significantly decreased post-occlusion renal pHi values. Thus, the inhibition of NHE3 with S3226 may be beneficial in treatment of ischemic ARF.