Nonlinear Full Invariant of Compact Banach-Space Maps

Abstract

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 (l 1 , . 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 uniformtopology subseries-convergent is that (X, . ) and (X , . ) both contain no copy of c 0 . This necessary condition is not sufficient.