The term “Holomorphic Spaces,” the title of a program held at the Mathematical Sciences Research Institute in the fall semester of 1995, is short for “Spaces of Holomorphic Functions.” It refers not so much to a branch of mathematics as to a common thread running through much of modern analysis—through functional analysis, operator theory, harmonic analysis, and, of course, complex analysis. This article will briefly outline the development of the subject from its origins in the early 1900's to the present, with a bias toward operator-theoretic aspects, in keeping with the main emphasis of the MSRI program. I hope that the article will be accessible not only to workers in the field but to analysts in general.

Origins

The subject began with the thesis of P. Fatou [1906], a student of H. Lebesgue. The thesis is a study of the boundary behavior of certain harmonic functions in the unit disk (those representable as Poisson integrals). It contains a proof, for example, that a bounded holomorphic function in the disk has a nontangential limit at almost every point of the unit circle. This initial link between function theory on the circle (real analysis) and function theory in the disk (complex analysis) recurred continually in the ensuing years. Some of the highlights are the paper of F. Riesz and M. Riesz [1916] on the absolute continuity of analytic measures; F.