Remarks on completely continuous polynomials

Abstract

In this paper the authors give new conditions on Banach spaces E and F which ensure that all polynomials P:E→F are completely continuous (i.e., send weakly converging sequences into norm-converging sequences). Among them are the following: (i) E has the Dunford-Pettis property and any Dunford-Pettis subset of F is relatively compact; (ii) all weakly null sequences of E are limited and F has the Gelʹfand-Phillips property. These results complement similar ones by M. González and J. M. Gutiérrez del Alamo [Arch. Math. (Basel) 63 (1994), no. 2, 145–151;Glasgow Math. J. 37 (1995), no. 2, 211–219;].
It is also shown that complete continuity of all polynomials in the Taylor expansion at a of a holomorphic function f does not imply the complete continuity of f, but that such a condition is equivalent to the "local'' complete continuity of f at a: Whenever a sequence (xn) converges weakly to a and ∥xn−a∥≤c<r for all n (here r is the radius of uniform convergence of f at a), then f(xn) is norm-convergent.