Compositions of projections in Banach spaces and relations between approximation properties

A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the π-property in terms of norms of compositions of projections is found. 2000 Mathematics Subject Classification. Primary 46B15; Secondary 46B07, 46B28. The problem of existence of Banach spaces with the π-property but without a finite dimensional decomposition is one of the well-known open problems in Banach space theory. It was first studied by W. B. Johnson [3]. P. G. Casazza and N. J. Kalton [2] found important connections of this problem with other problems of Banach space theory. See in this connection the survey [1]. Recall the definitions. A separable Banach space X has the π-property if there is a sequence Tn : X → X of finite dimensional projections such that (∀x ∈ X)( lim n→∞ ||x− Tnx|| = 0). If in addition the projections satisfy (∀n,m ∈ N)(TnTm = Tmin(m,n)), then X has a finite dimensional decomposition. Problem 1 Does every separable Banach space with the π-property have a finite dimensional decomposition? The purpose of this paper is to find an equivalent reformulation of Problem 1 in terms of norms of compositions of projections. In the second part of the paper we discuss related problems on compositions of projections.