Holomorphic Hilbert Bundles and the Frame Existence Problem

TL;DR

This paper demonstrates that certain Hilbert modules over a specific algebra lack frames, highlighting limitations in the existence of frames in the context of holomorphic Hilbert bundles.

Contribution

It introduces a counterexample of a Hilbert A-module without frames for the algebra A=K(l^2)+C·I_{l^2}, advancing understanding of the frame existence problem.

Findings

Existence of Hilbert A-modules without frames for A=K(l^2)+C·I_{l^2}

Counterexample impacts the theory of holomorphic Hilbert bundles

Highlights limitations in frame theory for certain operator algebras

Abstract

We show that if $A=K(l^2)+ \mathbb{C}I_{l^2}$, then there exists a Hilbert $A$-module that possess no frames.