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In this paper we ask when it is possible to transform a given sequence into aframe or a lower semi frame by multiplying the elements by numbers. In otherwords, we ask when a given sequence is a weighted frame or a weighted lowersemi frame and for each case we formulate a conjecture. We determine severalconditions under which these conjectures are true. Finally, we prove anequivalence between two older conjectures, the first one being that anyunconditionally convergent multiplier can be written as a multiplier of Besselsequences by shifting of weights, and the second one that every unconditionallyconvergent multiplier which is invertible can be written as a multiplier offrames by shifting of weights. We also show that these conjectures are alsorelated to one of the newly posed conjectures.

Weighted frames, weighted lower semi frames and unconditionally convergent multipliers