**ABSTRACT:**

It is known that a Bessel multiplier with a bounded weight is unconditionally convergent. In this talk we consider the converse situation. There are examples of unconditionally convergent multipliers, where at least one of the sequences is not Bessel or the weight is not bounded. In all these known examples, moving part of the weight to the sequences lead to a Bessel multiplier with a bounded weight. These examples open the question:

"If a multiplier is unconditionally convergent, can it be written as a Bessel multiplier with a bounded weight by moving part of the weight to the sequences?"

In this talk we determine classes of multipliers, where the answer of the question is "Yes". We pose the open problem to prove that the answer is always "Yes" or to find a counterexample.

The talk is based on a joint work with Peter Balazs.