Abstract:

An operator T : X → Y between Banach spaces is finitely strictly singular (FSS) if for every real a > 0 there exists a natural number n such that the unit sphere of every n-dimensional subspace of X contains a vector x such that ||Tx|| < a. The class of all FSS operators from X to Y is a subspace of L(X,Y); it contains the space of all compact operators and is contained in the space of all strictly singular operators from X to Y.

We will discuss FSS operators from ell_p to ell_q and from J_p to J_q with p < q, where J_p is the James' p-space. In particular, the formal inclusion operator is FSS in the both cases. For the former case, this was proved by Milman in 1970. The proof of the latter case is based on the following observation: every subspace of R^n of sufficiently large dimension contains a vector whose coordinates "strongly oscillate".

It was proved by Aronszajn and Smith in 1954 that every compact operator on a Banach space has an invariant subspace. In 1999, Read constructed a strictly singular operator without invariant subspaces. Since the class of FSS operators is intermediate between compact operators and strictly singular operators, it was of interest whether every FSS operator has an invariant subspace. We answer this question in the negative by proving that Read's operator is, actually, not just strictly singular but FSS.