Weighted Frames, Weighted Lower Semi Frames and Unconditionally Convergent Multipliers

In this paper, we ask when it is possible to transform a given sequence into a frame or a lower semi frame by multiplying the elements by numbers. In other words, we ask when a given sequence is a weighted frame or a weighted lower semi frame and for each case we formulate a conjecture. We determine several conditions under which these conjectures are true. Finally, we prove an equivalence between two older conjectures. The one being that any unconditionally convergent multiplier can be written as a multiplier of Bessel sequences by shifting weights between the generating sequences of the multiplier. The second one that every unconditionally convergent multiplier which is invertible can be written as a multiplier of frames by a similar shift of weights. We also show that these conjectures are also related to one of the newly posed conjectures.