Blueprint:
s-Numbers of Bounded Linear Operators
between Banach Spaces

3.5 Hilbert numbers \(h_n\)

For the Hilbert numbers we work over \(\mathbb {R}\) and use \(L_2 := \ell _2(\mathbb {N}; \mathbb {R})\).

Definition 3.11 Hilbert number
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Uses

Lean declarations

For \(S \in \mathcal{L}(X, Y)\) and \(n \in \mathbb {N}_0\),

\[ h_n(S) := \sup \biggl\{ \frac{a_n(B \, S \, A)}{\| B\| \, \| A\| } : A : L_2 \to X, \ B : Y \to L_2, \ A \neq 0, \ B \neq 0 \biggr\} . \]
Proposition 3.12
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Uses

Lean declarations

Let \(X, Y\) be Banach spaces. Then the Hilbert numbers \(h = (h_n)_n\) form an s-number sequence: they satisfy the axioms (S1)–(S5). The sequence \(h\) is not strict: (S5\('\)) fails in general.

Proof ▼

Each axiom is obtained by pushing the corresponding property of the approximation numbers \(a_n\) through the supremum defining \(h_n\): nonnegativity, antitonicity, subadditivity (S2), the ideal property (S3) and the rank axiom (S4) transfer directly. For the normalisations, \(h_0(S) = \| S\| \) combines the upper bound \(h_n(S) \le \| S\| \) with a rank-one factorisation of arbitrary unit vectors through \(L_2\), and \(h_n(\mathrm{id}_{\ell _2^{n+1}}) = 1\) (S5) tests the supremum with the canonical embedding and projection between \(\ell _2^{n+1}\) and \(L_2\).