Images of elliptic operators on Hilbert bundles on compact manifolds are closed

TL;DR

This paper proves that the image of elliptic operators on Hilbert bundles over compact manifolds is closed, facilitating the development of Hodge theory for such structures.

Contribution

It establishes the closedness of elliptic operator images on Hilbert bundles and compares tensor product images, advancing Hodge theory in this context.

Findings

Image of elliptic operators on Hilbert bundles is closed.

Tensor product of the operator preserves invariance and relates to the original image.

Provides groundwork for Hodge theory on Hilbert bundle structures.

Abstract

We prove that the image of an elliptic operator on a smooth separable Hilbert fibre bundle on compact manifolds is closed with respect to the natural pre-Hilbert topology. We consider a tensor product of the operator, which is invariant with respect to an action of the C*-algebra of compact operators, and compare the image of this tensor product with the image of the original operator. This establishes a ground for the Hodge theory for these structures.