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.
D. Alpay, V. Bolotnikov, A. Dijksma and J. Rovnyak. Some extensions of Loewner's theory of monotone operator functions. Journal of Functional Analysis Vol. 189 (2002), 1-20
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