A class of Banach spaces with few non-strictly singular operators

We construct a family (Xγ) of reflexive Banach spaces with long (countable as well as uncountable) transfinite bases but with no unconditional basic sequences. The method we introduce to achieve this allows us to considerably control the structure of subspaces of the resulting spaces as well as to precisely describe the corresponding spaces on non-strictly singular operators. For example, for every pair of countable ordinals γ,β, we are able to decompose every bounded linear operator from Xγ to Xβ as the sum of a diagonal operator and an strictly singular operator. We also show that every finite-dimensional subspace of any member Xγ of our class can be moved by and (4+ɛ)-isomorphism to essentially any region of any other member Xδ or our class. Finally, we find subspaces X of Xγ such that the operator space L(X,Xγ) is quite rich but any bounded operator T from X into X is a strictly singular pertubation of a scalar multiple of the identity.