Non-approximable compact operators

Abstract

Enflo’s negative answer to the approximation problem implies that there exist compact operators between Banach spaces that cannot be approximated by finite rank operators. Unfortunately, the standard approach is indirect and, hence, concrete examples are not provided. This unpleasant situation can be improved. Using infinite matrices constructed by Davie, the second-named author showed in his book “Operator Ideals” (proof of § 10.4.6) that the central part C _T of the canonical factorization

$$T : \ell_1 \stackrel{Q_T}{\longrightarrow} \, \ell_1/ \mathcal{N} (T) \stackrel{C_T}{\longrightarrow} \, \overline{\mathcal {M} (T)}\, \stackrel{J_T}{\longrightarrow}\, c_0$$

of certain operators ${T: \ell_1 \to c_0}$ is compact, but non-approximable. Though we have a good understanding of those operators ${T: \ell_1 \to c_0}$, they are still non-concrete since their generating matrix is obtained by stochastic methods. Looking at the central part of operators has far-reaching consequences: The ideal

$$\mathfrak{L}_q^{\rm ent} := \left\{T : \sum_{n = 1}^\infty e_n (T)^q < \infty \right\} \quad {\rm with} \quad 2 < q < \infty,$$

which is associated to the entropy numbers, contains non-approximable operators. As already shown in the (unpublished) thesis of the first-named author, the same conclusions hold for the ideals ${\mathfrak{L}_q^{\rm gel}}$ and ${\mathfrak{L}_q^{\rm kol}}$ generated by the Gelfand and Kolmogorov numbers, respectively. We present here a proof based on the new technique. The crux is the construction of non-approximable operators that are not just compact, but have a prescribed (not too strong) degree of compactness.