Minimal realizations of scalar generalized Nevanlinna functions related to their basic factorization

In this paper we present a minimal realization of a scalar generalized Nevanlinna function q which corresponds to the basic factorization of q as a product of a Nevanlinna function q(0) and of a rational function r(#)r, which collects the generalized poles and generalized zeros of q that are not of positive type. The key tool are reproducing kernel Pontryagin spaces.</p

Self-adjoint operators with inner singularities and pontryagin spaces

Let A(0) be an unbounded self-adjoint operator in a Hilbert space H-0 and let chi be a generalized element of order -m -1 in the rigging associated with Ag and the inner product (., .)(0) of H-0. In [S1, S2, S3] operators H-t, t epsilon R U {infinity}, are defined which serve as an interpretation for the family of operators A(0) + t(-1)(. , chi)(0) chi. The second summand here contains the inner singularity mentioned in the title. The operators H-t act in Pontryagin spaces of the form Pi(m) = H(0)circle plus C-m circle plus C-m where the direct summand space C-m circle plus C-m is provided with an indefinite inner product. They can be interpreted both as a canonical extension of some symmetric operator S in Pi(m) and also as extensions of a one-dimensional restriction S-0 of A(0) in H-0 and hence they can be characterized by a class of Straus extensions of S-0 as well as via M.G. Krein's formulas for (generalized) resolvents. In this paper we describe both these realizations explicitly and study their spectral properties. A main role is played by a special class of Q-functions. Factorizations of these functions correspond to the separation of the nonpositive type spectrum from the positive spectrum of H-t. As a consequence, in Subsection 7.3 a family of self-adjoint Hilbert space operators is obtained which can serve as a nontrivial quantum model associated with the operators Ag + t(-1)(. , chi)(0) chi.</p

Schur analysis in the quaternionic setting: The fueter regular and the slice regular case

This chapter is a survey on recent developments in quaternionic Schur analysis. The first part is based on functions which are slice hyperholomorphic in the unit ball of the quaternions, and have modulus bounded by 1. These functions, which by analogy to the complex case are called Schur multipliers, are shown to be (as in the complex case) the source of a wide range of problems of general interest. They also suggest new problems in quaternionic operator theory, especially in the setting of indefinite inner product spaces. This chapter gives an overview on rational functions and their realizations, on the Hardy space of the unit ball, on the half-space of quaternions with positive real part, and on Schur multipliers, also discussing related results. For the purpose of comparison this chapter presents also another approach to Schur analysis in the quaternionic setting, in the framework of Fueter series. To ease the presentation most of the chapter is written for the scalar case, but the reader should be aware that the appropriate setting is often that of vector-valued functions