7.3 The maximal difference theorem
The maximal difference theorem \(a_n \le e\, (n+1)\, h_n\) is due to Ullrich [ 12 ] . It is driven by the determinant quantities of Definition 7.8, which are squeezed between the two quantities of interest: they do not decay faster than the approximation numbers allow (Lemma 7.11, via the explicit rank-\(k\) approximant \(L = SA(BSA)^{-1}BS\)), and consecutive ratios are bounded by the Hilbert numbers (Lemma 7.12, via contractive compressions).
Call a pair \((A, B)\) admissible for \(S \in \mathcal{L}(X, Y)\) and \(k \in \mathbb {N}_0\) if \(A : \ell _2^k \to X\) and \(B : Y \to \ell _2^k\) are contractions. For such a pair \(B \circ S \circ A\) is an endomorphism of the \(k\)-dimensional Hilbert space \(\ell _2^k\), so its determinant is a scalar, and
One has \(\Delta _0(S) = 1\) (the determinant on the trivial space is \(1\)) and \(\Delta _k(S) \le \| S\| ^k\), via \(\prod _i a_i(T) = |\det T|\) (Lemma 7.9) and \(a_i(T) \le \| T\| \le \| S\| \).
For an endomorphism \(T\) of a nonzero finite-dimensional inner-product space, \(\prod _{k {\lt} \dim } a_k(T) = |\det T|\).
The approximation numbers coincide with the singular values \(\sigma _k(T)\) (Theorem 6.11), and \(|\det T| = \prod _k \sigma _k(T)\) is linear algebra: \(|\det T|^2 = \det (T^* T) = \prod _k \lambda _k(T^* T) = \prod _k \sigma _k(T)^2\).
The bordered-determinant identity is the elementary column-operation form of the Schur determinant formula; it needs no invertibility of the top-left block.
Let \(R\) be a commutative ring, \(M\) a \((k+1) \times (k+1)\) matrix over \(R\) and \(w \in R^k\) such that, above the corner, the last column of \(M\) is the combination of the first \(k\) columns with coefficients \(w\), i.e. \(M_{i,\mathrm{last}} = \sum _j M_{ij} w_j\) for \(i {\lt} k\). Then
where \(M'\) is the top-left \(k \times k\) block of \(M\).
Subtracting from the last column the combination of the other columns with coefficients \(w\) leaves the determinant unchanged and clears the column above the corner; Laplace expansion along it gives the claim.
For every \(S \in \mathcal{L}(X, Y)\) and \(k \in \mathbb {N}_0\),
Fix an admissible pair \((A, B)\) for \(\Delta _k(S)\) and put \(T := BSA\); if \(\det T = 0\) there is nothing to prove. Otherwise \(T\) is invertible, so \(L := SA\, T^{-1}BS\) factors through \(\ell _2^k\), hence has rank at most \(k\), hence \(\| S - L\| \ge a_k(S)\) by the definition of the approximation numbers. Pick \(x \in B_X\) with \(\| (S-L)x\| \) almost \(\| S - L\| \) and a Hahn–Banach functional \(b \in B_{Y^*}\) norming \((S-L)x\), and border the pair by them: \(A_0(\xi \oplus \tau ) := \lambda \, A\xi + \mu \, \tau \, x\) and \(B_0 y := (\lambda \, B y) \oplus (\mu \, b(y))\) with \(\lambda ^2 + \mu ^2 = 1\). Contractivity of \(A_0\) is the two-term Cauchy–Schwarz inequality \(\lambda a + \mu b \le \sqrt{a^2+b^2}\), and \(\| B_0 y\| ^2 = \lambda ^2\| By\| ^2 + \mu ^2 |b(y)|^2 \le \| y\| ^2\). Above the corner, the last column of the matrix of \(B_0SA_0\) is the combination of the first \(k\) columns with coefficients \((\mu /\lambda )\, \zeta \), where \(\zeta := T^{-1}B(Sx)\), so Lemma 7.10 gives
The weights \(\lambda ^2 = k/(k+1)\), \(\mu ^2 = 1/(k+1)\) maximise \(\lambda ^{2k}\mu ^2\), whose value is then exactly \(k^k/(k+1)^{k+1}\); taking the supremum over \((A, B)\) finishes. Note that no near-maximiser has to be selected: every admissible pair is improved.
In particular, if \(a_n(S) {\gt} 0\) then \(\Delta _k(S) {\gt} 0\) for all \(k \le n+1\), by induction from \(\Delta _0(S) = 1\) and \(a_k(S) \ge a_n(S)\).
\(\Delta _{n+1}(S) \le h_n(S)\, \Delta _n(S)\).
For admissible \((A, B)\) and \(T := B S A\) on \(\ell _2^{n+1}\), \(|\det T| = \prod _{k \le n} a_k(T)\) (Lemma 7.9). The last factor satisfies \(a_n(T) \le \| B\| \| A\| \, h_n(S) \le h_n(S)\). For the top \(n\) factors, the diagonal factorisation \(B_1 T A_1 = \operatorname {diag}(a_0(T), \dots , a_n(T))\) through contractions (Theorem 6.7), compressed to the first \(n\) coordinates by the coordinate embedding/projection, is itself admissible for \(\Delta _n(S)\), whence \(\prod _{k {\lt} n} a_k(T) \le \Delta _n(S)\). Again every admissible pair is bounded, so the bound passes to the supremum \(\Delta _{n+1}(S)\).
For every \(S \in \mathcal{L}(X, Y)\) and \(n \in \mathbb {N}_0\),
If \(a_n(S) = 0\) there is nothing to prove. Otherwise chain the growth lemma (Lemma 7.11) at \(k = n\) with the upper bound (Lemma 7.12): \(\frac{n^n}{(n+1)^{n+1}}\, a_n(S)\, \Delta _n(S) \le \Delta _{n+1}(S) \le h_n(S)\, \Delta _n(S)\), and cancel \(\Delta _n(S) {\gt} 0\). Finally \((n+1)^{n+1}/n^n = (n+1)(1 + 1/n)^n \le e\, (n+1)\) from \(1 + x \le e^x\) at \(x = 1/n\).
Since the Hilbert numbers are the smallest and the approximation numbers the largest s-number sequence, the theorem bounds any s-number sequence by any other: the maximal possible difference between two s-number sequences is a factor linear in \(n\). This is the conjecture of Carl and Pietsch [ 1 ] , proved up to the constant \(e\) in [ 12 ] .
Let \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \). For all s-number sequences \(s, t\), every \(S \in \mathcal{L}(X, Y)\) and every \(n \in \mathbb {N}_0\),
Taking \(t = b\) and \(s = c\) resp. \(s = d\) gives the Mityagin–Henkin conjecture [ 6 ] , again up to the constant \(e\).
Let \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \). For every \(S \in \mathcal{L}(X, Y)\) and every \(n \in \mathbb {N}_0\),
Apply Corollary 7.14 with \(t := b\) and \(s := c\) resp. \(s := d\) (both are s-number sequences) and take the maximum.
The factor \(n+1\) is order-optimal: the inclusion \(I \colon \ell _1 \to \ell _\infty \) of Section 8.3 satisfies \(c_n(I) \ge \tfrac 12\) while \(h_n(I) = (n+1)^{-1}\), so no bound \(a_n \le \varphi (n)\, h_n\) can hold with \(\varphi (n) = o(n)\) (Corollary 8.15).