Conservative Realizations of Herglotz-Nevanlinna Functions

This book is devoted to conservative realizations of various classes of Stieltjes, inverse Stieltjes, and general Herglotz-Nevanlinna functions as impedance functions of linear systems. The main feature of the monograph is a new approach to the realization theory profoundly involving developed extension theory in triplets of rigged Hilbert spaces and unbounded operators as state-space operators of linear systems. The connections of the realization theory to systems with accretive, sectorial, and contractive state-space operators as well as to the Phillips-Kato sectorial extension problem, the Krein-von Neumann and Friedrichs extremal extensions are provided. Among other results the book contains applications to the inverse problems for linear systems with non-self-adjoint Schrödinger operators, Jacobi matrices, and to the Nevanlinna-Pick system interpolation.

Preface. - 1 Extensions of Symmetric Operators. - 2 Rigged Hilbert Spaces. - 3 Bi-extensions of Closed Symmetric Operators. -. 4 Quasi-self-adjoint Extensions. - 5 The Livsic Canonical Systems with Bounded Operators. - 6 Herglotz-Nevanlinna functions and Rigged Canonical Systems. - 7 Classes of realizable Herglotz-Nevanlinna functions. - 8 Normalized Canonical Systems. - 9 Canonical L-systems with Contractive and Accretive Operators. - 10 Systems with Schrödinger operator. - 11 Non-self-adjoint Jacobi Matrices and System Interpolation. - 12 Non-canonical Systems. - Notes and Comments. - References. - Index.