8.3.1 Ingredients
Two inputs are needed for the Hilbert numbers of \(I\). The first pair of lemmas bounds a Hilbert–Schmidt-type sum by an operator norm; the second is an estimate on the leading singular values of a product.
Let \(w_j\) (\(j \in J\), \(J\) finite) be vectors in an inner product space and \(M \in \mathbb {R}\). If \(\| \sum _{j \in J} \varepsilon _j w_j\| \le M\) for every choice of signs \(\varepsilon \in \{ \pm 1\} ^J\), then \(\sum _{j \in J} \| w_j\| ^2 \le M^2\).
Summing over all \(2^{|J|}\) sign patterns, the cross terms cancel:
proved by induction on \(J\), pairing the two extensions of each sign pattern to a new index by the parallelogram law. Each summand on the left is at most \(M^2\), and there are \(2^{|J|}\) of them.
Let \(H\) be a Hilbert space and \(J\) a finite set of indices.
For every \(B \in \mathcal{L}(\ell _\infty , H)\), \(\displaystyle \sum _{j \in J} \| B e_j\| ^2 \le \| B\| ^2\).
For every \(A \in \mathcal{L}(H, \ell _1)\) with rows \(w_j \in H\), i.e. \(\langle w_j, x\rangle = (A x)_j\) for all \(x\), \(\displaystyle \sum _{j \in J} \| w_j\| ^2 \le \| A\| ^2\).
In Hilbert–Schmidt language: both operators are Hilbert–Schmidt with \(\| \cdot \| _{HS} \le \| \cdot \| \).
(1) The vectors \(\sum _{j \in J} \varepsilon _j e_j\) all have \(\ell _\infty \)-norm \(1\), so \(\| \sum _j \varepsilon _j B e_j\| \le \| B\| \) and Lemma 8.7 applies with \(M = \| B\| \).
(2) For a signed sum \(z := \sum _{j \in J}\varepsilon _j w_j\) the rows reproduce the signs on the coordinates of \(A z\):
so \(\| z\| \le \| A\| \), and Lemma 8.7 applies with \(M = \| A\| \). This is the little Grothendieck bound for the dual operator \(\ell _\infty = \ell _1^{\, \prime } \to H\), formulated through the rows so that no adjoint has to be constructed.
The second ingredient is a consequence of the Schmidt representation (Theorem 6.5): the quantitative form of the classical statement that the nuclear norm of a product is at most the product of the Hilbert–Schmidt norms of its factors [ 8 , 2.11.23 ] . The two Hilbert–Schmidt bounds enter as hypotheses, so neither the Hilbert–Schmidt nor the nuclear norm has to be defined.
Let \(S \in \mathcal{L}(H_1, H_2)\) be described by rows \(w_j \in H_1\), in the sense that \(\| S x\| ^2 = \sum _j |\langle w_j, x\rangle |^2\) for all \(x\), and suppose \(\sum _{j \in J} \| w_j\| ^2 \le c\) for every finite \(J\). Then \(\sum _{k \in s} \| S u_k\| ^2 \le c\) for every finite \(s\) and every family \((u_k)\) of pairwise orthogonal vectors of norm \(1\) or \(0\).
Interchange the two summations and apply Bessel’s inequality \(\sum _k |\langle u_k, w_j\rangle |^2 \le \| w_j\| ^2\) for each fixed \(j\), then sum over \(j\) and use the row bound.
Let \(T = T_2 T_1\) be compact, with \(T_1 \in \mathcal{L}(H_1, H_2)\) and \(T_2 \in \mathcal{L}(H_2, H_3)\), and let \(a, b \ge 0\) be such that
for all finite \(s\) and all orthonormal(-or-zero) families \((u_k)\) in \(H_1\) and \((v_k)\) in \(H_3\). Then
Take a Schmidt representation of \(T\) with singular values \(\sigma _k\) and singular vectors \(u_k, v_k\); by Eckart–Young \(\sigma _n = a_n(T)\), and since \(\sigma \) is antitone, \((n+1)\sigma _n \le \sum _{k \le n} \sigma _k\). For each \(k\) with \(\sigma _k \ne 0\) the vector \(v_k\) is a unit vector and \(T u_k = \sigma _k v_k\), whence
Summing over \(k \le n\) and applying Cauchy–Schwarz together with the two hypotheses gives \(\sum _{k \le n}\sigma _k \le b\, a\).