The foundations of this outstanding book series were laid in 1944. Until the end of the 1970s, a total of 77 volumes appeared, including works of such distinguished mathematicians as Carathéodory, Nevanlinna and Shafarevich, to name a few. The series came to its name and present appearance in the 1980s. In keeping its well-established tradition, only monographs of excellent quality are published in this collection. Comprehensive, in-depth treatments of areas of current interest are presented to a readership ranging from graduate students to professional mathematicians. Concrete examples and applications both within and beyond the immediate domain of mathematics illustrate the import and consequences of the theory under discussion.

The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint exit space extensions of a, not necessarily densely defined, symmetric operator, in terms of maximal dissipative (in $\dC_+$) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of a boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct the generalized resolvents from the given parameter family. The general version of the coupling method is introduced and the role of boundary relations and their Weyl families for the Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page

A nonnegative form t on a complex linear space is decomposed with respect to another nonnegative form w: it has a Lebesgue decomposition into an almost dominated form and a singular form. The part which is almost dominated is the largest form majorized by t which is almost dominated by w. The construction of the Lebesgue decomposition only involves notions from the complex linear space. An important ingredient in the construction is the new concept of the parallel sum of forms. By means of Hilbert space techniques the almost dominated and the singular parts are identified with the regular and a singular parts of the form. This decomposition addresses a problem posed by B. Simon. The Lebesgue decomposition of a pair of finite measures corresponds to the present decomposition of the forms which are induced by the measures. T. Ando's decomposition of a nonnegative bounded linear operator in a Hilbert space with respect to another nonnegative bounded linear operator is a consequence. It is shown that the decomposition of positive definite kernels involving families of forms also belongs to the present context. The Lebesgue decomposition is an example of a Lebesgue type decomposition, i.e., any decomposition into an almost dominated and a singular part. There is a necessary and sufficient condition for a Lebesgue type decomposition to be unique. This condition is inspired by the work of Ando concerning uniqueness questions.

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This paper consists of two chapters. The first chapter concerns matrix functions belonging to the generalized Nevanlinna class N:' * . We present results about the operator representation of such functions.These representations are then used to obtain information about the (generalized) poles of generalized Nevanlinna functions. The second chapter may be viewed as a continuation of our paper [DLS3] and treats Hamiltonian systems of differential equations with boundary conditions depending on the eigenvalue parameter. In particular we study the eigenvalues, both isolated and embedded eigenvalues. 1 of the equation (O.l)a and have certain properties which we repeat in Section 6 below. Recall that in [DLS3] with the problem (O.l)o,b there was associated a selfadjoint relation A with nonempty resolvent set p ( A ) in some Pontryagin or even Krein space R, which is an extension of the closure S of the symmetric minimal relation associated with the equation J f ' -H f = dg on (a, b) in the Hilbert space L2(d dt) (in the first part denoted by Lj(a,b)). In this paper we restrict ourselves to the case where R is a Pontryagin space. Then the selfadjoint relation A, if it is chosen minimal, is uniquely determined up to an isomorphism. We call it the linearization of the boundary eigenvalue problem (O.l)a,b and denote it by &.( Y d We acknowledge the support of the Netherlands organization for scientific research NWO.Chapter I. NZ functions 1. Representations of NE functionsAn m x m matrix function Q belongs to the (generalized) Neuanlinna class N; ", where K is a nonnegative integer, if it is defined and locally meromorphic on an open subset of C containing C\R, such that Q(O* = Q ( 0 and the kernel has K negative squares. We denote by GQ, 99 the largest open sets in C on which Q is locally meromorphic and locally holomorphic, respectively. Clearly, we have g Q c G9.An m x m matrix function K(l, I ) defined for I, I in some set 59, like NQ(l, I ) on gQ, is said to have K negative squares, if K(2, I)* = K(1, 0, 1, I E 9, and if for any choice of the number n E IN, the points l,, 1, . . . , 1, E 9 and the vectors cl, c,, . . . , c, E C" the Hermitian n x n matrix ((cYK(1, lj) ci))Zj= has at most K and at least for one of these choices precisely K negative eigenvalues. We write N, for Ni I. admits a representation of the formwhere A is a selfadjoint relation in some Pontryagin space n of index IC' 2 K, which has a nonempty resolvent set p ( A ) , p~p ( A ) f l C' is a fixed point of reference, ( A -0-I is the resolvent operator of A in 17 and r is a linear mapping from C" to n, cf. [KL2]. In Sections 2 and 3 we construct some models for 17, A and r for which (1.1) is valid. Clearly, it follows from this formula that p ( A ) c 9@Simple examples show that there exist selfadjoint relations in a Pontryagin space of which the resolvent sets are empty (cf. [DS2]), but the resolvent set of a (densely defined) selfadjoint operator in a Pontryagin space is automatically nonempty, see [IKL].Without loss of generality we may and shall a...