The kernel bundle of a holomorphic Fredholm family

TL;DR

This paper constructs a smooth vector bundle from a family of holomorphic Fredholm operators, linking kernel classes to meromorphic elements, with applications in boundary value problems for elliptic wedge operators.

Contribution

It introduces a method to associate a smooth kernel bundle to a holomorphic Fredholm family, expanding the understanding of boundary value problems in elliptic wedge operators.

Findings

Constructed a smooth vector bundle of kernel classes.

Established a link between meromorphic elements and kernel classes.

Provided examples relevant to boundary value problems for elliptic wedge operators.

Abstract

Let $\Y$ be a smooth connected manifold, $\Sigma\subset\C$ an open set and $(\sigma,y)\to\scrP_y(\sigma)$ a family of unbounded Fredholm operators $D\subset H_1\to H_2$ of index 0 depending smoothly on $(y,\sigma)\in \Y\times \Sigma$ and holomorphically on $\sigma$. We show how to associate to $\scrP$, under mild hypotheses, a smooth vector bundle $\kerb\to\Y$ whose fiber over a given $y\in \Y$ consists of classes, modulo holomorphic elements, of meromorphic elements $\phi$ with $\scrP_y\phi$ holomorphic. As applications we give two examples relevant in the general theory of boundary value problems for elliptic wedge operators.