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Theorems due to Stenger (Bull Am Math Soc 74:369–372, 1968) and Nudelman (Int Equ Oper Theory 70:301–305, 2011) in Hilbert spaces and their generalizations to Krein spaces in Azizov and Dijksma (Int Equ Oper Theory 74(2):259–269, 2012) and Azizov et al. (Linear Algebra Appl 439:771–792, 2013) generate additional questions about properties a finite-codimensional compression $${T_0}$$ of a symmetric or self-adjoint linear relation $${T}$$ may or may not inherit from $${T}$$ . These questions concern existence of invariant maximal nonnegative subspaces, definitizability, singular critical points and defect indices.

For plus-operators in a Banach indefinite space, we consider a linear-fractional relation. The classes of operators with the empty domain of definition for such a relation are described. The sufficient (and necessary, in some meaning) conditions for the chain rule to be valid are given.

For closed linear operators or relations A and B acting between Hilbert spaces \({\mathcal{H}}\) and \({\mathcal{K}}\) the concepts of compact and finite rank perturbations are defined with the help of the orthogonal projections PA and PB in \({\mathcal{H}} \oplus {\mathcal{K}}\) onto the graphs of A and B. Various equivalent characterizations for such perturbations are proved and it is shown that these notions are a natural generalization of the usual concepts of compact and finite rank perturbations.

It was shown by P. Jonas and H. Langer that a selfadjoint definitizable operator A in a Krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator B is selfadjoint and the resolvent set ρ(B) is nonempty. It is the aim of this note to prove a more general variant of this perturbation result where the assumption on ρ(B) is dropped. As an application a class of singular ordinary differential operators with indefinite weight functions is studied.

Via approximative eigensequences we introduce the notion of spectral points of type π+ and π- for self-adjoint operators in Krein spaces. They are stable under compact perturbations. For real spectral points of type π+ and π- which are not in the interior of the spectrum we prove that the growth of the resolvent in some neighbourhood of them is of finite order. There exists a local spectral function with singularities. It turns out that all spectral subspaces corresponding to sufficiently small neighbourhoods of points of type π+ or type π- are Pontryagin spaces.

We give necessary and sufficient conditions under which a C-0-semigroup of bi-contractions on a Krein space is similar to a semigroup of contractions on a Hilbert space. Under these and additional conditions we obtain direct sum decompositions of the Krein space into invariant regular subspaces and we describe the behavior of the semigroup on each of these summands. In the last section we give sufficient conditions for the co-generator of the semigroup to be power bounded. (C) 2004 Elsevier Inc. All rights reserved.

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.

Certain meromorphic matrix valued functions on C\R; the so-called boundary coefficients, are characterized in terms of a standard symmetric operator S in a Pontryagin space with finite (not necessarily equal) defect numbers, a meromorphic mapping into the defect subspaces of S; and a boundary mapping for S: Under some simple assumptions the boundary coefficients also satisfy a minimality condition. It is shown that these assumptions hold if and only if for S a generalized von Neumann equality is valid. r 2002 Elsevier Science (USA). All rights reserved. MSC: primary 47B50; 47B25; 34B07; 47B32; secondary 46C20; 47A06

in some Hilbert space. The operator matrix is a maximal uniformly accretive operator which is selfadjoint in this space with respect to some indefinite metric. The theorem on strong solvability of the hydrodynamic problem is proved. Further, the spectrum of normal oscillations, basis properties of eigenfunctions and other questions are studied.

A prominent class of -symmetric Hamiltonians is for some nonnegative number N. The associated eigenvalue problem is defined on a contour Γ in a specific area in the complex plane (Stokes wedges), see [3, 5]. In this short note we consider the case N = 2 only. Here we elaborate the relationship between Stokes lines and Stokes wedges and well-known limit point/limit circle criteria from [6,10,11]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

In the recent years a generalization $H=p^2 +x^2(ix)^\epsilon$ of the harmonic oscillator using a complex deformation was investigated, where \epsilon\ is a real parameter. Here, we will consider the most simple case: \epsilon even and x real. We will give a complete characterization of three different classes of operators associated with the differential expression H: The class of all self-adjoint (Hermitian) operators, the class of all PT symmetric operators and the class of all P-self-adjoint operators. Surprisingly, some of the PT symmetric operators associated to this expression have no resolvent set.

Let \( \mathcal{H} \) be a J-space and let \( V = \left( {\begin{array}{*{20}c} {V_1 } & {V_{12} } \\ {V_{21} } & {V_2 } \\ \end{array} } \right) \) be the matrix representation of a J-binoncontractive operator V with respect to the canonical decomposition \( \mathcal{H} = \mathcal{H}^ + \oplus \mathcal{H}^ - \) of \( \mathcal{H} \). The main aim of this paper is to show that the assumption $$ V_{12} (V_2 - V_{21} V_1^{ - 1} V_{12} ) \in \mathfrak{S}_\infty $$ (0.1) implies the existence of a V-invariant maximal nonnegative subspace. Let us note that (0.1) is a generalization of the well-known M.G. Krein condition \( V_{12} \in \mathfrak{S}_\infty \). The set of all operators satisfying (0.1) is described via Potapov-Ginsburg transform.

Let S and T be linear operators in a linear space such that S ⊂ T. In this note an estimate for the codimension of dom S n in dom T n in terms of the codimension of dom S in dom T is obtained. An immediate consequence is that for any polynomial p the operator p(S) is a finite-dimensional restriction of the operator p(T) whenever S is a finite-dimensional restriction of T. The general results are applied to a perturbation problem of self-adjoint definitizable operators in Krein spaces.

The aim of this paper is to study the completeness and basicity problems for selfadjoint operators of the class K(H) in almost Krein spaces and prove criteria for the basicity and completeness of root vectors of linear pencils.

Given two monic polynomials P 2 n and P 2 n − 2 of degree 2n and 2 n − 2 ( n ⩾ 2 ) with complex coefficients and with disjoint zero sets. We give necessary and sufficient conditions on these polynomials such that there exist two n × n Jacobi matrices B and C for which P 2 n ( λ ) = det ( λ 2 I n + λ B + C ) , P 2 n − 2 ( λ ) = det ( λ 2 I n − 1 + λ B 1 + C 1 ) , where B 1 and C 1 are the ( n − 1 ) × ( n − 1 ) Jacobi matrices obtained from B and C by deleting the last row and the last column. The zeros of P 2 n and P 2 n − 2 are the eigenvalues of the quadratic Jacobi matrix pencils on the right-hand side of the equalities, whence the title of the paper. The problem is formulated and solved in a slightly more general form.

We extend the notion of definitizability from selfadjoint operators in Krein spaces to products of selfadjoint operators in Hilbert spaces and show that the spectral properties essentially remain the same. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

The Schur algorithm as developed by C. Chamfy [C] and J. Dufresnoy [D] (see also [BDGPS]) is related to a sequence of characteristic functions of closely outerconnected coisometric colligations in Pontryagin spaces.

Consider an operator pencil A0+λ1A1+···+λnAn in which, for example (other cases are also considered), A0 is a maximal accretive operator, A1, ..., An are closed accretive operators, and dom A0 ⊂ dom Aj, j = ¯1,n¯. We give a sufficient condition under which it is closed for all λj ≥ 0, j = ¯1,n¯. In case n = 1, domA0 = domA1, and A0, A1 are maximal uniformly accretive, this condition is also necessary. The condition is that the matrix (cos(Ai,Aj))ni,j=0 is uniformly cone positive. Here cos(Ai,Aj) is the cosine of the angle between Ai and Aj. We prove some new and reprove some old results related to uniform cone positivity and the cosine. In the final section we study the closedness of some 2 × 2 matrices with operator entries.

In the recent years a generalization of Hermiticity was investigated using a complex deformation H=p^2 +x^2(ix)^\epsilon of the harmonic oscillator Hamiltonian, where \epsilon is a real parameter. These complex Hamiltonians, possessing PT symmetry (the product of parity and time reversal), can have real spectrum. We will consider the most simple case: \epsilon even. In this paper we describe all self-adjoint (Hermitian) and at the same time PT symmetric operators associated to H=p^2 +x^2(ix)^\epsilon. Surprisingly it turns out that there are a large class of self-adjoint operators associated to H=p^2 +x^2(ix)^\epsilon which are not PT symmetric.

In the lifting theorem of B. Sz.-Nagy, C. Foias, and D. Sarason the intertwining operator A, the operator that is lifted, is a contraction from one Hilbert space to another. J. A. Ball and J. W. Helton showed that a kind of lifting theorem still holds if the negative spectral subspace of the operator I-A*A is finite dimensional. In this paper we prove that this result remains valid if the Hilbert spaces are replaced by spaces with an indefinite metric. (C) 1997 Academic Press.