Some extensions of Loewner's theory of monotone operator functions

Several extensions of Loewner's theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class, The theory of monotone operator functions is generalized from scalar- to matrix-valued functions of an operator argument. A notion of kappa-monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented. (C) 2002 Elsevier Science (USA).</p

Some Extensions of Loewner\u27s Theory of Monotone Operator Functions

Several extensions of Loewner’s theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar- to matrix-valued functions of an operator argument. A notion of -monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented

Notes on Interpolation in the Generalized Schur Class. II. Nudelman\u27s Problem

An indefinite generalization of Nudel′man’s problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides results on existence criteria for Pick-Nevanlinna and Carath´eodory-Fej´er interpolation, the method yields new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel′man’s problem

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

AbstractWe prove that a 2×2 matrix polynomial which is J-unitary on the real line can be written as a product of normalized elementary J-unitary factors and a J-unitary constant. In the second part we give an algorithm for this factorization using an analog of the Schur transformation