On the minimal Sums of sequences in the tensor product of separable Hilbert spaces

TL;DR

This paper generalizes the characterization of frames in tensor products of Hilbert spaces to finite minimal sums of tensor product sequences, providing new insights into Bessel sequences and Gabor systems.

Contribution

It introduces a framework for understanding when finite sums of tensor product sequences form Bessel sequences or frames, extending previous results.

Findings

S is a Bessel sequence iff each term is a tensor product of Bessel sequences.

Necessary conditions for S to be a frame are established.

Results on Gabor systems generated by finite rank functions are derived.

Abstract

It is known that the tensor product of two sequences, in the tensor product of two separable Hilbert spaces, is a frame if and only if each component of that product is a frame. This paper proposes a sort of generalization of the aforementioned result by dealing with sequences S that are finite minimal sums of tensor products of a finite number of sequences. We prove that S is a Bessel sequence if and only if it is a sum for which each term is the tensor product of Bessel sequences. We also state necessary conditions for S to be a frame. For dimensions higher than one, we deduce several results on Gabor systems generated by finite rank square integrable functions. Meanwhile, the one dimensional versions of some of these results are surprisingly extremely difficult to prove or disapprove.