s-Numbers: Dependencies

Let \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \). For all s-number sequences \(s, t\), every \(S \in \mathcal{L}(X, Y)\) and every \(n \in \mathbb {N}_0\),

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

Let \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \). For every \(S \in \mathcal{L}(X, Y)\) and every \(n \in \mathbb {N}_0\),

\(h_n(I) = (n+1)^{-1}\) for all \(n\).

For all \(n\),

whereas the maximal difference theorem gives \(a_n \le e\, (n+1)\, h_n\) in general. Hence no bound \(a_n \le \varphi (n)\, h_n\) can hold with \(\varphi (n) = o(n)\).

Let \(s\) be an s-number sequence. For every \(S \in \mathcal{L}(X, Y)\) and \(n \in \mathbb {N}_0\),

A family \(s = (s_n)_{n \in \mathbb {N}_0}\) as in Definition 2.1 is additive if for all normed spaces \(X, Y\), all \(S, T \in \mathcal{L}(X, Y)\) and all \(n, m \in \mathbb {N}_0\),

\(S \in \mathcal{L}(X, Y)\) is approximable if \(a_n(S) \xrightarrow {n \to \infty } 0\), equivalently if \(S\) is the operator-norm limit of a sequence of finite-rank operators.

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

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

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

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

where \(B_X\) is the closed unit ball of \(X\) and the centres of the covering balls are arbitrary points of \(Y\). (The indexing is \(0\)-based, so \(e_0\) allows a single ball.)

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

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

A family \(s\) as in Definition 2.1 is injective if post-composition with any metric injection leaves it unchanged, \(s_n(J \circ S) = s_n(S)\); it is surjective if pre-composition with any metric surjection leaves it unchanged, \(s_n(S \circ Q) = s_n(S)\).

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

where \(\pi _V : Y \to Y/V\) is the quotient map and \(Y/V\) carries the quotient seminorm.

A metric injection is a continuous linear map \(J : Y \to Z\) that preserves norms: \(\| J y\| = \| y\| \) for all \(y\) (an isometric embedding). In particular \(\| J\| \le 1\).

A metric surjection is a continuous linear map \(Q : W \to X\) that is a contraction (\(\| Q\| \le 1\)) and maps the open unit ball onto the open unit ball: every \(x\) has preimages of norm arbitrarily close to \(\| x\| \). This is the quotient-map condition \(\| x\| = \inf \{ \| w\| : Q w = x\} \).

A family \(s = (s_n)_{n \in \mathbb {N}_0}\) as in Definition 2.1 is multiplicative if for all normed spaces \(X, Y, Z\), all \(T \in \mathcal{L}(X, Y)\), \(S \in \mathcal{L}(Y, Z)\) and all \(n, m \in \mathbb {N}_0\),

A family \(s = (s_n)_{n \in \mathbb {N}_0}\) as in Definition 2.1 is a pseudo-s-number sequence if it satisfies (S1), (S2) and (S3) [ 8 , 12.1.1 ] . Every s-number sequence is one (IsSNumberSequence.isPseudoSNumberSequence).

For \(S \in \mathcal{L}(X, Y)\) the rank of \(S\) is the cardinal dimension of its range,

A family

defined for all pairs of normed spaces \((X, Y)\) over \(\mathbb {K}\) is called an s-number sequence if:

(S1)

\(\| S\| = s_0(S) \ge s_1(S) \ge \dots \ge 0\) for every \(S\).

(S2)

\(s_n(S + T) \le s_n(S) + \| T\| \).

(S3)

\(s_n(BSA) \le \| B\| \, s_n(S) \, \| A\| \) for composable bounded operators \(A, B\).

(S4)

\(s_n(S) = 0\) whenever \(\operatorname {rank}(S) \le n\).

(S5)

\(s_n(\mathrm{id}_{\ell _2^{n+1}}) = 1\).

An s-number sequence is strict if it additionally satisfies the strengthening

(S5\('\))

\(s_n(\mathrm{id}_X) = 1\) for every normed space \(X\) with \(\dim X {\gt} n\), not merely on \(X = \ell _2^{n+1}\).

\(J_Y\) is a metric injection: \(\| J_Y y\| = \| y\| \) for every \(y \in Y\).

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 \(M\).

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

\(\Delta _{n+1}(S) \le h_n(S)\, \Delta _n(S)\).

Let \(H\) be an inner-product space and \(J\) a finite set of indices.

1. For every \(B \in \mathcal{L}(\ell _\infty , H)\), \(\displaystyle \sum _{j \in J} \| B e_j\| ^2 \le \| B\| ^2\).

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 \(\| \cdot \| _{HS} \le \| \cdot \| \).

Post-composition with a metric injection and pre-composition with a metric surjection both leave the operator norm unchanged: \(\| J \circ T\| = \| T\| \) and \(\| T \circ Q\| = \| T\| \).

For an endomorphism \(T\) of a nonzero finite-dimensional inner-product space, \(\prod _{k {\lt} \dim } a_k(T) = |\det T|\).

Let \(S \in \mathcal{L}(H_1, H_2)\) be described by rows \(w_j \in H_1\), \(j \in \mathbb {N}_0\), 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 \(J \subseteq \mathbb {N}_0\). 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 \(0\).

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 \(\sum _{j \in J} \| w_j\| ^2 \le M^2\).

For every s-number sequence \(s\), all \(S, T \in \mathcal{L}(X, Y)\) and every \(n \in \mathbb {N}_0\),

The approximable operators form a closed two-sided operator ideal: a subspace of \(\mathcal{L}(X, Y)\), closed under pre/post-composition with bounded operators, closed in operator-norm topology, and containing all finite-rank operators.

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

If \(\dim X {\gt} n\) then \(e_n(\mathrm{id}_X) \ge 1/2\).

If \(S(B_X)\) contains more than \(2^n\) points that are pairwise at distance at least \(\delta \), then \(e_n(S) \ge \delta / 2\).

Let \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \). For every \(S \in \mathcal{L}(X, Y)\), \(e_0(S) = \| S\| \).

The Hilbert numbers \(h = (h_n)_n\) form an s-number sequence that is additive: they satisfy the axioms (S1)–(S5), and for all \(S, T \in \mathcal{L}(X, Y)\) and all \(n, m \in \mathbb {N}_0\),

For all \(n\), \(\tfrac 12 \le c_n(I) \le 1\).

\(h_n(I) \ge (n+1)^{-1}\) for all \(n\).

For every strict s-number sequence \(s\), every \(1 \le p \le \infty \) and \(n {\lt} m\), we have \(s_n(\mathrm{id}_{\ell _p^m}) = 1\).

On a Hilbert space, every s-number sequence agrees with the approximation numbers: \(s_n(S) = a_n(S)\) for every continuous linear map \(S\) between Hilbert spaces.

The approximation numbers \(a = (a_n)_n\) are additive and multiplicative: for all \(n, m \in \mathbb {N}_0\),

If \(\mathbb {K}\) is complete, they form a strict s-number sequence: they satisfy (S1)–(S5) and the strengthening (S5\('\)).

For \(S \in \mathcal{L}(X, Y)\) between Banach spaces, \(a_n(S) \le (1 + \sqrt{n})\, c_n(S)\).

For \(S \in \mathcal{L}(X, Y)\) between Banach spaces, \(a_n(S) \le (1 + \sqrt{n})\, d_n(S)\).

For \(S \in \mathcal{L}(X, Y)\) between Banach spaces,

Every approximable operator \(S : X \to Y\) from a normed space \(X\) to a Banach space \(Y\) over a locally compact field \(\mathbb {K}\) is compact. In particular every operator of finite rank is compact, since finite-rank operators are approximable (Proposition 6.2).

Let \(\mathbb {K}\) be a non-trivially normed field and \(X, Y\) normed spaces over \(\mathbb {K}\). Then the Bernstein numbers \(b = (b_n)_n\) form a strict s-number sequence: they satisfy (S1)–(S5) and the strengthening (S5\('\)). If \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \), then \(b\) is neither additive nor multiplicative.

On Hilbert spaces, \(S\) is approximable iff \(S\) is compact.

For \(S \in \mathcal{L}(X, Y)\), each of \(d_n(S) \to 0\) and \(c_n(S) \to 0\) is equivalent to total boundedness of \(S(B_X)\). If \(Y\) is complete, then

Let \(H_1, H_2\) be Hilbert spaces and \(s\) any s-number sequence. Then \(S \in \mathcal{L}(H_1, H_2)\) is compact if and only if \(s_n(S) \to 0\).

Every compact \(S \in \mathcal{L}(H_1, H_2)\) between Hilbert spaces is approximable.

In John position, that is, when the identity is a maximal-volume feasible operator for the body seminorm \(q\), the identity is a positive combination of the rank-one projections onto contact points:

where a contact point is a unit vector \(u\) with \(\operatorname {Re}\langle x, u\rangle \le q(x)\) for all \(x\).

Let \(S \in \mathcal{L}(X, Y)\) and suppose every \(x \in B_X\) satisfies \(\| Sx - y_i\| \le \varepsilon \) for some member of a family \(y_1, \dots , y_k \in Y\). Then

Let \(1 \le p {\lt} q {\lt} \infty \) and define \(r\) by \(\tfrac 1r = \tfrac 1p - \tfrac 1q\). For \(D_\sigma : \ell _q^m \to \ell _p^m\),

Let \(1 \le q \le p \le \infty \) (including \(p = \infty \)). For \(D_\sigma : \ell _q^m \to \ell _p^m\),

Let \(1 \le p \le \infty \) and \(D_\sigma : \ell _p^m \to \ell _p^m\). Then \(\| D_\sigma \| = \max _i |\sigma _i|\), and for every strict s-number sequence \(s\) (in particular for \(a_n\), \(b_n\), \(c_n\), \(d_n\)),

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

For every \(S \in \mathcal{L}(X, Y)\), the sequence \(e_n(S)\) tends to \(0\) if and only if \(S(B_X)\) is totally bounded. If \(Y\) is complete, then

where the forward implication needs no completeness.

Let the norm values of \(\mathbb {K}\) be dense in \([0, \infty )\) (as for \(\mathbb {R}\) and \(\mathbb {C}\)). Then for all \(S \in \mathcal{L}(X, Y)\), \(B \in \mathcal{L}(Y, Z)\) and \(n, m \in \mathbb {N}_0\),

Let \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \). For every \(S \in \mathcal{L}(X, Y)\) with \(S \neq 0\) and every \(n \in \mathbb {N}_0\), \(e_n(S) {\gt} 0\). In particular (S4) fails for every nonzero operator of finite rank.

Let \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \). The entropy numbers \(e = (e_n)_n\) form a pseudo-s-number sequence: they satisfy (S1), (S2) and (S3).

For all normed spaces \(X\), \(Y\) over \(\mathbb {K}\), every \(S \in \mathcal{L}(X, Y)\) and every \(n \in \mathbb {N}_0\),

The Gelfand numbers \(c_n\) form an injective s-number sequence: \(c_n(J \circ S) = c_n(S)\) for every metric injection \(J\).

Let \(\mathbb {K}\) be a non-trivially normed field and \(X, Y, Z\) normed spaces over \(\mathbb {K}\). Then the Gelfand numbers \(c = (c_n)_n\) form a strict s-number sequence that is additive and multiplicative: they satisfy (S1)–(S5) and the strengthening (S5\('\)), and for all \(n, m \in \mathbb {N}_0\),

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

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

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

Let \(\mathbb {K}\) be a complete densely normed field. For every Banach space \(X\) and every \(S \in \mathcal{L}(X, Y)\), the canonical Kolmogorov number equals the approximation number of the lift: \(d_n(S) = a_n(S \circ Q_X)\). Completeness of \(\mathbb {K}\) is needed: over \(\mathbb {Q}\), with \(X = Y = \mathbb {R}\) and \(S = \mathrm{id}\), one has \(d_1(S) = 0\) because \(\mathbb {Q}\) is dense in \(\mathbb {R}\), while \(a_1(Q_X) = 1\).

The Kolmogorov numbers \(d = (d_n)_n\) are additive and multiplicative: for all \(n, m \in \mathbb {N}_0\),

If \(\mathbb {K}\) is complete, they form a strict s-number sequence: they satisfy (S1)–(S5) and the strengthening (S5\('\)).

Let \(\mathbb {K}\) be a complete densely normed field. For operators whose domains are Banach spaces, \(d^Q\) satisfies (S1)–(S4), and \(d^Q_n(\mathrm{id}_X) = 1\) for every Banach space \(X\) with \(\dim X {\gt} n\).

The Kolmogorov numbers are surjective: \(d_n(S \circ Q) = d_n(S)\) for every metric surjection \(Q\).

\(h_n(I) \le (n+1)^{-1}\) for all \(n\).

For a body seminorm \(p\) on \(\mathbb {K}^k\) equivalent to the Euclidean norm, among the feasible operators (those \(T\) with \(p(Tu) \le \| u\| \) for all \(u\)) there is one, \(T_0\), of maximal \(|\det |\), and it is invertible (\(\det T_0 \ne 0\)).

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

For \(1 \le p \le q {\lt} \infty \) and \(m \ge 1\),

Let \(H_1, H_2 \ne \{ 0\} \). Every compact \(S \in \mathcal{L}(H_1, H_2)\) attains its operator norm as a singular value: there are unit vectors \(u, v\) with \(S u = \| S\| \, v\) and \(S^* v = \| S\| \, u\).

Let \(X\) be a normed \(\mathbb {K}\)-space and \(M \subseteq X\) a closed subspace of codimension \(\le n\). Then for every \(\varepsilon {\gt} 0\) there is a bounded projection \(P : X \to X\) (\(P \circ P = P\)) with \(\ker P = M\) and \(\| P\| \le \sqrt{n} + \varepsilon \).

Let \(Y\) be a normed \(\mathbb {K}\)-space and \(V \subseteq Y\) a subspace of dimension \(\le n\). Then there is a bounded projection \(P : Y \to Y\) (\(P \circ P = P\)) with \(\operatorname {range} P = V\) and \(\| P\| \le \sqrt{n}\).

For every s-number sequence \(s\) and every operator \(S\),

For any bounded \(S \in \mathcal{L}(H_1, H_2)\) between Hilbert spaces and any real \(c\) with \(0 \le c {\lt} a_n(S)\), there are contractions \(A \in \mathcal{L}(\ell _2^{n+1}, H_1)\), \(B \in \mathcal{L}(H_2, \ell _2^{n+1})\) with \(B \circ S \circ A = c \cdot \mathrm{id}_{\ell _2^{n+1}}\).

Let \(H_1, H_2\) be finite-dimensional Hilbert spaces over \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \). For every s-number sequence \(s\), every \(S \in \mathcal{L}(H_1, H_2)\) and every \(n \in \mathbb {N}_0\), we have

where \(\sigma _n(S)\) are Mathlib’s eigenvalue-defined singular values.

For \(1 \le p \le q {\lt} \infty \) and \(n {\lt} m\),

Let \(H_1, H_2 \ne \{ 0\} \). For every compact \(S \in \mathcal{L}(H_1, H_2)\) and every \(n \in \mathbb {N}_0\) there exist contractions \(A \in \mathcal{L}(\ell _2^{n+1}, H_1)\) and \(B \in \mathcal{L}(H_2, \ell _2^{n+1})\) such that, on every basis vector \(e_k\) of \(\ell _2^{n+1}\),

Equivalently, \(B \circ S \circ A\) is the diagonal operator \(\operatorname {diag}(a_0(S), \dots , a_n(S))\) on \(\ell _2^{n+1}\).

For every compact \(S \in \mathcal{L}(H_1, H_2)\) there exist sequences \((u_k) \subseteq H_1\), \((v_k) \subseteq H_2\), each orthonormal after its zero terms are removed, with \(u_k, v_k \ne 0\) whenever \(a_k(S) {\gt} 0\), such that

More precisely, every such expansion with nonincreasing nonnegative coefficients \(\sigma _k\) has \(\sigma _k = a_k(S)\).

Let \(\gamma {\lt} d_n(S)\). Then there are \(x_0, \dots , x_n \in B_X\) and subspaces \(V_0, \dots , V_n \subseteq Y\) with \(S x_k \in V_j\) for \(k {\lt} j\) and \(\| S x_j + w\| {\gt} \gamma \) for all \(w \in V_j\). Dually, for \(0 {\lt} \gamma {\lt} c_n(S)\) there are \(x_0, \dots , x_n \in B_X\) and functionals \(\rho _0, \dots , \rho _n\) with \(\| \rho _j\| \le 1\), \(\rho _j(S x_j) = \gamma \) and \(\rho _i(S x_j) = 0\) for \(i {\lt} j\).

Let \(H_1, H_2 \ne \{ 0\} \). For every compact \(S \in \mathcal{L}(H_1, H_2)\) and every \(n \in \mathbb {N}_0\) there exists \(L \in \mathcal{L}(H_1, H_2)\) with \(\operatorname {rank}(L) \le n\) and \(\| S - L\| = a_n(S)\).

Every finite-dimensional normed space over \(\mathbb {R}\) admits a basis \((e_i)\) with \(\| e_i\| = 1\) whose dual coordinate functionals also satisfy \(\| e_i^*\| = 1\).