Operators in the Cowen-Douglas class and related topics

Abstract

Linear spaces with an Euclidean metric are ubiquitous in mathematics, arising both from quadratic forms and inner products. Operators on such spaces also occur naturally. In recent years, the study of multivariate operator theory has made substantial progress. Although the study of self-adjoint operators goes back a few decades, the non self-adjoint theory has developed at a slower pace. While several approaches to this topic has been developed, the one that has been most fruitful is clearly the study of Hilbert spaces that are modules over natural function algebras like $\mathcal A({\Omega})$, where $\Omega \subseteq \mathbb C^m$ is a bounded domain, consisting of complex valued functions which are holomorphic on some open set $U$ containing $\overline{\Omega}$, the closure of $\Omega$. The book, ''Hilbert Modules over function algebra'', R. G. Douglas and V. I. Paulsen showed how to recast many of the familiar theorems of operator theory in the language of Hilbert modules. The book, ''Spectral decomposition of analytic sheaves'', J. Eschmeier and M. Putinar and the book, ''Analytic Hilbert modules'', X. Chen and K. Guo, provide an account of the achievements from the recent past. The impetus for much of what is described below comes from the interplay of operator theory with other areas of mathematics like complex geometry and representation theory of locally compact groups.