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...