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 \((A_0, A_1).\) Lions problem, posed in 1960s, is to prove that different parameters \((\theta ,p)\) produce different interpolation spaces \((A_0, A_1)_{\theta , p}.\) We first establish conditions on \(A_0\) and \(A_1\) so that interpolation spaces of this couple are strictly intermediate spaces between \(A_0+A_1\) and \(A_0\cap A_1.\) 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 \(p\ne p^*\) the operator ideals \({\mathcal {L}}^{(a)}_{p,q}(X,Y),\) \({\mathcal {L}}^{(a)}_{p^*,q^*}(X,Y)\) generated by approximation numbers are distinct. Moreover, for any fixed p,  either all operator ideals \({\mathcal {L}}^{(a)}_{p,q}(X,Y)\) 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 \({\mathcal {L}}^{(s)}_{p,q}(X,Y)\) is strictly embedded into \({\mathcal {L}}^{\mathcal {A}}(X,Y).\)