Diversity of Lorentz-Zygmund Spaces of Operators Defined by Approximation Numbers

We prove the following dichotomy for the spaces ℒ ^(a)_p,q,α (X, Y) of all operators T ∈ ℒ(X, Y) whose approximation numbers belong to the Lorentz-Zygmund sequence spaces ℓ_p,q(log ℓ)_α: If X and Y are infinite-dimensional Banach spaces, then the spaces ℒ ^(a)_p,q,α (X, Y) with 0 < p < ∞, 0 < q ≤ ∞ and α ∈ ℝ are all different from each other, but otherwise, if X or Y are finite-dimensional, they are all equal (to ℒ(X, Y)).

Moreover we show that the scale \({\{ {\cal L}_{\infty ,q}^{(a)}(X,Y)\} _{0\, < q\, < \infty }}\) is strictly increasing in q, where ℒ ^(a)_∈,q (X, Y) is the space of all operators in ℒ(X, Y) whose approximation numbers are in the limiting Lorentz sequence space ∓_∈,q.