Nonlinear Full Invariant of Compact Banach-Space Maps

We characterize a nonlinear full invariant of compact Banach-space maps: Let (X, ‖.‖) and (Y, ‖.‖) be two Banach spaces and PC(X, Y) be all compact maps which map (X, ‖.‖) to (Y, ‖.‖). Then each weak operator-topology subseries-convergent series ∑iPi in Pc(X, Y) is also uniform-topology subseries-convergent iff each bounded map from (X, ‖.‖) to (l1, ‖.‖1) is a compact map. The necessary condition for each weak operator-topology subseries-convergent series ∑iPi in PC(X, Y) to be also uniform-topology subseries-convergent is that (X, ‖.‖) and (X′, ‖.‖) both contain no copy of c0. This necessary condition is not sufficient.