A note on compact operators on normed linear spaces

We show that there is no surjective compact operator on a normed linear infinite-dimensional space.     

Rank one perturbations at infinite coupling in Pontryagin spaces

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^∞.     

Pontryagin spaces of entire functions II

We continue the study of a generalization of L. de Branges's theory of Hilbert spaces of entire functions to the Pontryagin space setting. In this-second-part we investigate isometric embeddings of spaces of entire functions into spacesL ² () understood in a distributional sense and consider Weyl coefficients of matrix chains. The main task is to give a proof of an indefinite version of the inverse spectral theorem for Nevanlinna functions. Our methods use the theory developed by L. de Branges and the theory of extensions of symmetric operators of M.G.Krein.     

Projections in Krein spaces

Let (H,J) be a Krein space with selfadjoint involution J. Starting with a canonical representation of a J-selfadjoint projection, J-projection in short, as the sum of a J-positive projection and a J-negative one we study in detail the structure of a regular subspace, that is, the range of a J-projection. We treat the problem when the sum of two regular subspaces is again regular. We also treat the problem when the closure of the range of the product of a J-contraction and a J-expansion becomes regular.     

Hermitian indefinite functions and Pontryagin spaces of entire functions

We establish and investigate a connection between hermitian indefinite continuous functions with finitely many negative squares defined on a finite interval and so-called de Branges spaces of entire functions. This enables us to relate to any hermitian indefinite continuous function on the real axis a certain chain of 2×2-matrix valued entire functions, which are in the positive definite case tightly connected with canonical systems of differential equations.     

On von Neumann regular f-algebras

It is shown that for a large class of f-algebras, von Neumann regularity and -lateral completeness are equivalent notions.     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

Boundary Value Problems with Local Generalized Nevanlinna Functions in the Boundary Condition

For a class of abstract λ-dependent boundary value problems where a local variant of generalized Nevanlinna functions appears in the boundary condition, linearizations are constructed and their local spectral properties are investigated.     

Pontryagin spaces of entire functions I

We give a generalization of L.de Branges theory of Hilbert spaces of entire functions to the Pontryagin space setting. The aim of this-first-part is to provide some basic results and to investigate subspaces of Pontryagin spaces of entire functions. Our method makes strong use of L.de Branges's results and of the extension theory of symmetric operators as developed by M.G.Krein.     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Interpolation problems,extensions of symmetric operators and reproducing kernel spaces II

By an oversight on the part of the authors this section was not included in the paper previously published in Integral Equations Operator Theory, volume 14/4 (1991), 466–500. Present address:Department of Mathematics Ben-Gurion University of the Negev Beersheva Israel     

Toeplitz operators with noncommuting symbols

LetM be a von Neumann algebra with a faithful normal tracial state and letH be a finite maximal subdiagonal subalgebra ofM. LetH ² be the closure ofH in the noncommutative Lebesgue spaceL ²(M). We consider Toeplitz operators onH ² whose symbol belong toM, and find that they possess several of the properties of Toeplitz operators onH ²( ) with symbol fromL ( ), including norm estimates, a Hartman-Wintner spectral inclusion theorem, and a characterisation of the weak^* continuous linear functionals on the space of Toeplitz operators.     

Characterization of domains of self-adjoint ordinary differential operators

The GKN (Glazman, Krein, Naimark) Theorem characterizes all self-adjoint realizations of linear symmetric (formally self-adjoint) ordinary differential equations in terms of maximal domain functions. These functions depend on the coefficients and this dependence is implicit and complicated. In the regular case an explicit characterization in terms of two-point boundary conditions can be given. In the singular case when the deficiency index d is maximal the GKN characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter λ. In the much more difficult intermediate cases, not all solutions contribute to the singular self-adjoint conditions. In 1986 Sun found a representation of the self-adjoint singular conditions in terms of certain solutions for nonreal values of λ. In this paper we give a representation in terms of certain solutions for real λ. This leads to a classification of solutions as limit-point (LP) or limit-circle (LC) in analogy with the celebrated Weyl classification in the second-order case. The LC solutions contribute to the singular boundary conditions, the LP solutions do not. The advantage of using real λ is not only because it is, in general, easier to find explicit solutions but, more importantly, it yields information about the spectrum.     

Equilibrium existence theorems in KKM spaces

An abstract convex space satisfying the KKM principle is called a KKM space. This class of spaces contains G$G$-convex spaces properly. In this work, we show that a large number of results in KKM theory on G$G$-convex spaces also hold on KKM spaces. Examples of such results are theorems of Sperner and Alexandroff–Pasynkoff, Fan–Browder type fixed point theorems, Horvath type fixed point theorems, Ky Fan type minimax inequalities, variational inequalities, von Neumann type minimax theorems, Nash type equilibrium theorems, and Himmelberg type fixed point theorems.     

Norm-attaining compact operators

We show examples of compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. This is the negative answer to an open question posed in the 1970s. Actually, any strictly convex Banach space failing the approximation property serves as the range space. On the other hand, there are examples in which the domain space has a Schauder basis.     

The bitangential inverse spectral problem for canonical systems

The theory of the direct and bitangential inverse input impedance problem is used to solve the direct and bitangential inverse spectral problem. The analysis of the direct spectral problem uses and extends a number of results that appear in the literature. Special attention is paid to the class of canonical integral systems with matrizants that are strongly regular J-inner matrix valued functions in the sense introduced in [7]. The bitangential inverse spectral problem is solved in this class. In our considerations, the data for this inverse problem is a given nondecreasing p×p matrix valued function σ(μ) on and a normalized monotonic continuous chain of pairs , of entire inner p×p matrix valued functions. Each such chain defines a class of canonical integral systems in which we find a solution of the inverse problem for the given spectral function σ(μ). A detailed comparison of our investigations of inverse problems with those of Sakhnovich is presented.