Finitely strictly singular operators between James spaces

AbstractAn 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 dimE⩾n contains a vector x such that ‖Tx‖<ε‖x‖. We show that, for 1⩽p<q<∞, the formal inclusion operator from Jp to Jq 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 Rn contains a vector x with ‖x‖ℓ∞=1 such that xmi=(−1)i for some m1<⋯