Generalized Shifts on Cartesian Products

It is proved that if E, F are infinite dimensional strictly convex Banach spaces totally incomparable in a restricted sense, then the Cartesian product E×F with the sum or sup norm does not admit a forward shift. As a corollary it is deduced that there are no backward or forward shifts on the Cartesian product`p1×`p2,1\u3c p16=p2\u3c∞, with the supremum norm thus settling a problem left open in Rajagopalan and Sundaresan in J. Analysis 7 (1999(, 75-81 and also a problem stated as unsolved in Rassias and Sundaresan