The Schur Transformation for Nevanlinna Functions: Operator Representations, Resolvent Matrices, and Orthogonal Polynomials
نویسنده
چکیده
A Nevanlinna function is a function which is analytic in the open upper half plane and has a non-negative imaginary part there. In this paper we study a fractional linear transformation for a Nevanlinna function n with a suitable asymptotic expansion at ∞, that is an analogue of the Schur transformation for contractive analytic functions in the unit disc. Applying the transformation p times we find a Nevanlinna function np which is a fractional linear transformation of the given function n. The main results concern the effect of this transformation to the realizations of n and np, by which we mean their representations through resolvents of self-adjoint operators in Hilbert space. Our tools are block operator matrix representations, u–resolvent matrices, and reproducing kernel Hilbert spaces. Mathematics Subject Classification (2000). Primary 47A57, 47B32, Secondary 42C05, 30E05.
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