Finitely strictly singular operators between James spaces

Abstract

An operator T : X → Y between Banach spaces is said to be finitely strictly singular if for every ε > 0 there exists n such that every subspace E ⊆ X with dim E ⩾ n contains a vector x such that ‖ T x ‖ < ε ‖ x ‖ . We show that, for 1 ⩽ p < q < ∞ , the formal inclusion operator from J p to J q is finitely strictly singular. As a consequence, we obtain that the strictly singular operator with no invariant subspaces constructed by C. Read is actually finitely strictly singular. These results are deduced from the following fact: if k ⩽ n then every k-dimensional subspace of R n contains a vector x with ‖ x ‖ ℓ ∞ = 1 such that x m i = ( − 1 ) i for some m 1 < ⋯ < m k .