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 1960’s, 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 $p\mathrm{\ne }{p}^{\ast }$ the operator ideals ${\mathcal{L}}_{p,q}^{(a)}(X,Y)$, ${\mathcal{L}}_{p^{\ast },q^{\ast }}^{(a)}(X,Y)$ generated by approximation numbers are distinct. Moreover, for any fixed $p$, either all operator ideals ${\mathcal{L}}_{p,q}^{(a)}(X,Y)$ 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 ${\mathcal{L}}_{p,q}^{(s)}(X,Y)$ is strictly embedded into ${\mathcal{L}}^{\mathcal{A}}(X,Y)$. The paper is dedicated to the memory of Prof. A. Pietsch, who passed away recently.