Pick-Nevanlinna interpolation theorem and multiplication operators on functional Hilbert spaces (Q1191288)

In an attempt to understand the curvature inequalities, we were led to some of the topics discussed below. After recalling some of the basic definitions, we discuss the Pick-Nevanlinna Theorem on the disk in section 2. Our main result is that if we replace the Szegö kernel by any other positive kernel the Pick-Nevanlinna interpolation theorem will be false. In section 3, we discuss Abrahamse's generalization of the Pick-Nevanlinna interpolation theorem to a finitely connected domain in \(C\). Finally, in section 4, we prove that it is not possible to establish an analogue of the Pick-Nevanlinna interpolation theorem for the unit ball. In this connection, we wish to point out that, some recent results of Jim Agler overlap with the material in this paper.