Georgia Institute of Technology College of Sciences

An intesting class of bounded operators or algebras of bounded operators on Hilbert spaces, particularly on Hilbert spaces of holomorphic functions, have a natural interpretation in terms of concepts from complex geometry. In particular, there is an intrinsic hermitian holomorphic vector bundle and many questions can be answered in terms of the Chern connection and the associated curvature. In this talk we describe this setup and some of the results obtained in recent years using this approach. The emphasis will be on concrete examples, particularly in the case of Hilbert spaces of holomorphic functions such as the Hardy and Bergman spaces on the unit sphere in C^n.