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 in X which are weakly convergent to some e∈X and Q((ei)i∈N,e) is a coanalytic subset of XN. 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: if X is a Banach space, (aij)i,j∈N is a regular method of summability and (ei)i∈N is a bounded sequence in X, then there exists a subsequence (ei)i∈L such that either (I) there exists e∈X such that every subsequence (ei)i∈H of (ei)i∈L is weakly summable w.r.t. (aij)i,j∈N to e and Q((ei)i∈H,e); or (II) for every subsequence (ei)i∈H of (ei)i∈L and every e∈X with Q((ei)i∈H,e)the sequence (ei)i∈H is not weakly summable to e w.r.t. (aij)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.