7 Inequalities between s-numbers

Beyond the qualitative sandwich \(h_n(S) \le s_n(S) \le a_n(S)\) (Theorem 4.2), one can compare the individual example sequences quantitatively; see [ 9 ] for an overview. This chapter formalises two such comparisons for operators between Banach spaces over \(\mathbb {K}\):

• the forward bound \(a_n(S) \le (1+\sqrt{n})\min (c_n(S), d_n(S))\) — the largest s-number exceeds the Gelfand and Kolmogorov numbers by at most a factor \(1+\sqrt{n}\) (classical; see [ 6 ] and [ 8 , § 2.10 ] );

• the reverse bound, the maximal difference theorem \(a_n(S) \le e\, (n+1)\, h_n(S)\) — the maximal possible difference between the largest s-numbers, the approximation numbers, and the smallest ones, the Hilbert numbers, is a factor linear in \(n\) [ 10 ] . Its consequence for two arbitrary s-number sequences is Corollary 7.14, the conjecture of Carl and Pietsch [ 1 ] ; specialising to the Gelfand, Kolmogorov and Bernstein numbers gives Corollary 7.15, the conjecture of Mityagin and Henkin [ 5 ] — both up to the constant \(e\).

Everything in this chapter is proved in Inequalities.lean and MaxDifference.lean (the bordered-determinant lemma, a pure matrix fact, in Determinant.lean). The two classical projection theorems that drive the forward bound (Garling–Gordon and Kadets–Snobar) are proved, for \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \) at once, by the John’s-ellipsoid development of Section 7.1, whose analytic core is the John decomposition of identity (Theorem 7.2).