**Date & Time:** Tuesday, February 23, 2016, 15:30-16:30

**Venue:** Ramanujan Hall

**Title:** The role of curvature in operator theory

**Speaker: ** Prof. Gadadhar Misra, Mathematics Department, IISc Bangalore

**Abstract:** Starting with a positive definite kernel $K$ defined on a
bounded open connected subset $\Omega$ of $\mathbb C^d,$ we give several
canonical constructions for producing new positive definite kernels on
$\Omega,$ possibly taking values in $Hom(E)$ for some normed linear space
$E$ of dimension $d.$ Specifically, this includes the curvature defined
as the $d\times d$ matrix of real analytic functions
$$\big ( \!\! \big ( \tfrac{\partial}{\partial_i \bar{\partial}_j} \log K
\big ) \!\!\big ).$$
These kernels define an inner product on a submodule (over the polynomial
ring) functions holomorphic on $\Omega.$ The completion is a Hilbert space
on which the polynomials act by point-wise multiplication making it into a
"Hilbert module". We will discuss hereditary properties, sub and quotient
of these Hilbert modules.