Unconditionally convergent multipliers and Bessel sequences

TL;DR

This paper demonstrates that unconditionally summable sequences in Hilbert spaces can be factorized into scalar and Bessel sequences, providing insights into multipliers and confirming a conjecture in specific cases.

Contribution

It introduces a factorization result for unconditionally summable sequences and applies it to unconditionally convergent multipliers, addressing a conjecture by Balazs and Stoeva.

Findings

Unconditionally summable sequences can be factorized into scalar and Bessel sequences.

Positive results on representing unconditionally convergent multipliers.

Partial confirmation of Balazs and Stoeva's conjecture.

Abstract

We prove that every unconditionally summable sequence in a Hilbert space can be factorized as the product of a square summable scalar sequence and a Bessel sequence. Some consequences on the representation of unconditionally convergent multipliers are obtained, thus providing positive answers to a conjecture by Balazs and Stoeva in some particular cases.