Hibschweiler Rita A., MacGregor Thomas H. Banach Spaces of Analytic Functions

American Mathematical Society, 2008. — 160 p.Banach spaces of analytic functions play an important role in both classical and modern analysis. In this Proceedings, the major focus is on spaces of functions analytic in the open unit disc, such as the Hardy spaces, the Bergman spaces, and the weighted versions of these spaces. Other spaces under consideration here include the Bloch space, the families of Cauchy transforms and fractional Cauchy transforms, BMO, VMO, and the Fock space. Some of the work deals with questions in several complex variables.One of the themes of the Proceedings is the notion of multiplication operators, composition operators and weighted composition operators acting on Banach spaces. This has been a topic of extensive research over the past twenty years.
The Proceedings includes results characterizing bounded, compact and isometric composition operators in various settings.The recently published book "The Cauchy Transform" (2006) by Joseph A. Cima, Alec L. Matheson and William T. Ross, deals with material that is central to this Proceedings. A related book "Fractional Cauchy Transforms" (2006) was written by the editors of this Proceedings. Research by these authors is part of the Proceedings. The articles here were generated by the Special Session on Banach Spaces of Analytic Functions, held as part of the regional AMS meeting at the University of New Hampshire in April, 2006. It was very unfortunate that Alec Matheson, a major contributor to this field, died shortly before the meeting after a long illness. A brief note on the life of Alec Matheson, including several of his
results with various collaborators, was prepared by Joseph Cima and is included here.This volume presents both expository and research articles. Several authors conclude their articles with open questions.
The Special Session on Banach Spaces of Analytic Functions brought together thirty mathematicians, mostly from the northeastern United States. We hope that this volume serves to communicate more broadly the research presented at the meeting.