A dichotomy result for a pointwise summable sequence of operators

Let X be a separable Banach space and Q be a coanalytic subset of XN × X. We prove that the set of sequences(ei)i ∈ N inX which are weakly convergent to somee ∈ X andQ ((ei)i ∈ N, e) is a coanalytic subset ofXN . The proof applies methods of effective descriptive set theory to Banach space theory. Using Silver's Theorem [J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970) 60-64], this result leads to the following dichotomy theorem: ifX is a Banach space,(ai j)i, j ∈ N is a regular method of summability and(ei)i ∈ N is a bounded sequence inX, then there exists a subsequence(ei)i ∈ L such that either (I) there existse ∈ X such that every subsequence(ei)i ∈ H of(ei)i ∈ L is weakly summable w.r.t.(ai j)i, j ∈ N to e and Q ((ei)i ∈ H, e); or (II) for every subsequence(ei)i ∈ H of(ei)i ∈ L and everye ∈ X withQ ((ei)i ∈ H, e) the sequence(ei)i ∈ H is not weakly summable to e w.r.t. (ai j)i, j ∈ N. This is a version for weak convergence of an Erdös-Magidor result, see [P. Erdös, M. Magidor, A note on Regular Methods of Summability, Proc. Amer. Math. Soc. 59 (2) (1976) 232-234]. Both theorems obtain some considerable generalizations. © 2009 Elsevier B.V. All rights reserved