Polynomial convergence of sequences in Banach spaces.

Abstract

Let E be a (complex) Banach space and n be a positive integer, and denote by P( n E) the space of all n -homogeneous polynomials on E . The authors say that E has the m -FJ property (m -Farmer-Johnson property) if, whenever (x n ) n is a sequence in E which converges weakly to x in E and P(x n ) converges to P(x) for all P∈P( m E) , then Q(x n ) converges to Q(x) for all Q∈P( k E) for all k , 1≤k≤m . If all m -homogeneous polynomials are weakly sequentially continuous, then E has the m -FJ property. Using this observation, the authors give an alternative proof of the result that if E is a Banach space such that every m -homogeneous polynomial on E is weakly continuous on bounded sets then every k -homogeneous polynomial, 1≤k≤m , is weakly continuous on bounded sets [C. Boyd and R. A. Ryan, Arch. Math. (Basel) 71 (1998), no. 3, 211–218;]. It is shown that the m -FJ property is equivalent to the condition that, whenever y∈E and (x n ) n is a sequence in E which converges weakly to x in E and moreover P(x n ) converges to P(x) for all P∈P( m E) , then P(x n +y) converges to P(x+y) for all P∈P( m E) . The main result of the paper is the following: Let E be a Banach space with an unconditional decomposition E=∑ ∞ k=1 E k . Suppose each E k is a Banach space such that, if (x n ) n is a sequence in E k with the property that whenever x n converges weakly to x and P(x n ) converges to P(x) for all P∈P( m E k ) , then x k converges in norm to x . Then E has the m -FJ property. If J denotes the James space, a corollary to the main theorem is that the space (∑⊗J) l p , 1≤p<∞ , has the m -FJ property for every integer m .