Detailed characterization of unconditional convergence and invertibility of multipliers

In this paper we investigate the possibility of unconditional convergence and invertibility of multipliers $M_{m,\mathrm{\Phi },\mathrm{\Psi }}$ depending on the properties of the sequences $\mathrm{\Psi }$,$\mathrm{\Phi }$ and $m$. We characterize a complete set of conditions for the invertibility and the unconditional convergence of multipliers, and collect those results in tables. We either prove that unconditional convergence and invertibility is not possible, that one or both of these conditions are always the case for the given parameters, or we give examples for the feasible combinations. We give a full list of examples for all conditions.