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 P_C(X, Y) be all compact maps which map (X, ‖.‖) to (Y, ‖.‖). Then each weak operator-topology subseries-convergent series ∑_i P_i in P_c(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 ∑_i P_i in P_C(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.PACS: 02.10 By, 02.10 Gd