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.