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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.

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?

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.

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 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.

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∞.

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.

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

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.

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".

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.

An operator H = H0 +V where H0 = i (N is arbitrary) and V is a differential operator of order N-1 with coefficients decaying sufficiently rapidly at infinity is considered in the space H2(R). The goal of the paper is to find an expression for the trace of the difference of the resolvents (H) -1 and (H0 - z) -1 in terms of the Wronskian of appropriate solutions to the differential equation Hu = zu. This also leads to a representation for the perturbation determinant of the pair H0H.