  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.
