The Unconditional Constant for Hilbert Space Frame Expansions

We will prove some new results in frame theory by computing the unconditional constant (for all definitions of unconditional) for the frame expansion of a vector in a Hilbert space and see that it is $\sqrt{B/A}$, where $A,B$ are the frame bounds of the frame. It follows that tight frames have unconditional constant one. We then generalize this to a classification of such frames by showing that Bessel sequences have frame expansions with unconditional constant one if and only if the sequence is an orthogonal sum of tight frames. We give further results concerning frame expansions. We also give examples to show that our results are best possible. We end by discussing the connections of this work to {\it frame multipliers}. These results hold in both real and complex Hilbert spaces.