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Truncated moment and related interpolation problems in the class of generalized Nevanlinna functions are investigated. General solvability criteria will be established and complete parametrizations of solutions are given. The framework used in the paper allows the treatment of even and odd order problems in a parallel manner. The main new results concern the case where the corresponding Hankel matrix of moments is degenerate. One of the new effects in the indefinite case is that the degenerate moment problem may have infinitely many solutions. However, with a careful application of an indefinite analogue of a step-by-step Schur algorithm a complete description of the set of solutions will be obtained.

Boundary value problems for singular canonical systems of differential equations of the form Jf'(t) - H(t)f(t) = lambda Delta(t)f(t), t is an element of i, lambda is an element of C, are studied in the associated Hilbert space L(Delta)(2)(i). With the help of a monotonicity principle for matrix functions their square-integrable solutions are specified. This yields a direct treatment of defect numbers of the minimal relation and simultaneously makes it possible to assign certain boundary values to the elements of the maximal relation induced by the system of differential equations in L(Delta)(2)(i). The investigation of boundary value problems for these systems and their spectral theory can be carried out by means of abstract boundary triplet techniques. This paper makes explicit the construction and the properties of boundary triplets and Weyl functions for singular canonical systems. Furthermore, the Weyl functions are shown to have a property similar to that of the classical Titchmarsh-Weyl coefficients for singular Sturm-Liouville operators: they single out the square-integrable solutions of the homogeneous systems of canonical differential equations. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Let S be a closed symmetric operator with defect numbers (1, 1) in a Hilbert space h and let A be a selfadjoint operator extension of S in h. Then S is necessarily a graph restriction of A and the selfadjoint extensions of S can be considered as graph perturbations of A, cf. [8]. Only when S is not densely defined and, in particular, when S is bounded, S is given by a domain restriction of A and the graph perturbations reduce to rank one perturbations in the sense of [23]. This happens precisely when the Q-function of S and A belongs to the subclass N-0 of Nevanlinna functions. In this paper we show that by going beyond the Hilbert space h the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space h is given a one-dimensional extension, or the use of Hilbert space triplets associated with A is invoked. If the Q-function of S and A belongs to the subclass N-1 of Nevanlinna functions, then it is convenient to describe the selfadjoint extensions of S including its generalized Friedrichs extension (see [6]) by interpolating the original triplet, cf. [5]. For the case when A is semibounded, see also [4]. We prove some invariance properties, which imply that such an interpolation is independent of the (nonexceptional) extension.