A dichotomy result for a pointwise summable sequence of operators

Abstract

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