Using the Baire Category Theorem to Explore Lions Problem for Quasi-Banach Spaces

Many results for Banach spaces also hold for quasi-Banach spaces. One important such example is results depending on the Baire Category Theorem (BCT). We use the BCT to explore Lions problem for a quasi-Banach couple (A₀, A₁). Lions problem, posed in 1960's, is to prove that different parameters (θ,p) produce different interpolation spaces (A₀, A₁)θ, p. We first establish conditions on A₀ and A₁ so that interpolation spaces of this couple are strictly intermediate spaces between A₀+A₁ and A₀∩ A₁. This result, together with a reiteration theorem, gives a partial solution to Lions problem for quasi-Banach couples. We then apply our interpolation result to (partially) answer a question posed by Pietsch. More precisely, we show that if p≠ p^* the operator ideals L⁽ᵃ⁾p,q(X,Y), L⁽ᵃ⁾p^*,q^*(X,Y) generated by approximation numbers are distinct. Moreover, for any fixed p, either all operator ideals L⁽ᵃ⁾p,q(X,Y) collapse into a unique space or they are pairwise distinct. We cite counterexamples which show that using interpolation spaces is not appropriate to solve Pietsch's problem for operator ideals based on general s-numbers. However, the BCT can be used to prove a lethargy result for arbitrary s-numbers which guarantees that, under very minimal conditions on $X,Y$, the space L⁽ˢ⁾p,q(X,Y) is strictly embedded into LA(X,Y). The paper is dedicated to the memory of Prof. A. Pietsch, who passed away recently.