Compactness measured by s-numbers #
Which s-number sequences detect compactness? The answer separates the
sequences sharply, and this file collects the whole picture.
The general Banach case: cₙ and dₙ #
For the Gelfand and Kolmogorov numbers, decay to zero is equivalent to compactness over any Banach space:
S is a compact operator ↔ cₙ(S) → 0 ↔ dₙ(S) → 0.
Both proofs run through total boundedness of S(B_X), exactly as the entropy
criterion SNumbers.isCompactOperator_iff_tendsto_entropyNumber does.
- Compactness implies decay. A totally bounded image has, for every
ε, a finiteε-net; the covering estimates ofSNumbers.EntropyBoundsturn a net withkpoints into the boundsd_k(S) ≤ εandc_k(S) ≤ 2ε, and both sequences are antitone. - Decay implies compactness. For the Kolmogorov numbers,
dₙ(S) < εputsS(B_X)withinεof a bounded piece of a finite-dimensional subspace ofY, which is totally bounded. For the Gelfand numbers,cₙ(S) < εgives a closedM ⊆ Xof finite codimension with‖S|_M‖ < ε; the image ofB_Xin the finite-dimensional quotientX ⧸ Mis totally bounded, and lifting a finiteδ-net of it back toB_Xproduces a finite net forS(B_X).
The second half of the Gelfand argument deserves a comment, because the obvious
alternative fails. One would like to split x along X = M ⊕ F with F
finite-dimensional, but a projection with kernel M only satisfies
‖x - P x‖ ≤ (1 + √n)‖x‖ (Garling–Gordon), and (1 + √n)·cₙ(S) need not tend
to 0. Working in the quotient avoids any projection: a lifted difference
x - uⱼ is corrected by some m ∈ M with ‖m‖ ≤ 2 + δ, a bound independent
of the codimension n.
The approximation numbers are different #
aₙ(S) → 0 — that is, SVD.IsApproximable S — always implies compactness, but
the converse fails on a Banach space without the approximation property (Enflo,
1973): there are compact operators that are not norm limits of finite-rank
operators. On Hilbert spaces the Schmidt representation repairs this, and then
every s-number sequence detects compactness, since all of them agree with
aₙ there (SNumbers.allSNumbers_eq_on_HilbertSpace).
Main results #
SNumbers.isCompactOperator_iff_totallyBounded_image_closedBall— the bridge betweenIsCompactOperatorand total boundedness ofS(B_X).SNumbers.isCompactOperator_iff_tendsto_kolmogorovNumber,SNumbers.isCompactOperator_iff_tendsto_gelfandNumber— the two criteria.SNumbers.tendsto_kolmogorovNumber_iff_totallyBounded,SNumbers.tendsto_gelfandNumber_iff_totallyBounded— the same statements without any completeness assumption on the target space.SVD.IsCompactOperator.isApproximable,SVD.isApproximable_iff_isCompactOperator— compact ↔ approximable on Hilbert spaces.SVD.isCompactOperator_iff_tendsto_sn— on Hilbert spaces, compactness is equivalent tosₙ(S) → 0for everys-number sequences.
References #
Compactness and total boundedness #
A compact operator maps the closed unit ball to a totally bounded set.
Conversely, if S(B_X) is totally bounded and Y is complete, then S is
a compact operator: the closure of a totally bounded set is then compact.
For a complete target space, compactness of S is total boundedness of
S(B_X).
The Kolmogorov numbers #
If S(B_X) is totally bounded then dₙ(S) → 0: a finite ε/2-net with k
points bounds d_k(S) by ε/2, and dₙ is antitone.
If dₙ(S) → 0 then S(B_X) is totally bounded.
Given η > 0, choose V of dimension ≤ n with ‖π_V ∘ S‖ < η/3. Every
S x with ‖x‖ ≤ 1 is then within η/3 of a point of V of norm at most
‖S‖ + η/3; that bounded piece of the finite-dimensional space V is compact,
so a finite η/3-net of it turns into a finite η-net of S(B_X).
dₙ(S) → 0 characterises total boundedness of S(B_X). No completeness
assumption is needed.
A compact operator has Kolmogorov numbers tending to zero.
Kolmogorov numbers tending to zero force compactness, provided the target space is complete.
Compactness is measured by the Kolmogorov numbers: for a complete target
space, S is a compact operator if and only if dₙ(S) → 0.
The Gelfand numbers #
If S(B_X) is totally bounded then cₙ(S) → 0: a finite ε/3-net with k
points bounds c_k(S) by 2ε/3, and cₙ is antitone.
If cₙ(S) → 0 then S(B_X) is totally bounded.
Given η > 0, choose a closed M of finite codimension with ‖S|_M‖ < η/6.
The image of B_X in the finite-dimensional quotient X ⧸ M is bounded, hence
totally bounded; pick a finite δ-net of it inside the image and lift the
centres to u₁, …, u_N ∈ B_X. For ‖x‖ ≤ 1 with ‖[x] - [uⱼ]‖ < δ there is
m ∈ M with ‖(x - uⱼ) - m‖ < δ, and — this is the crux — ‖m‖ ≤ 2 + δ ≤ 3
independently of the codimension. Hence
‖S x - S uⱼ‖ ≤ ‖S m‖ + ‖S‖·δ < 3·(η/6) + η/2 = η.
cₙ(S) → 0 characterises total boundedness of S(B_X). No completeness
assumption is needed.
A compact operator has Gelfand numbers tending to zero.
Gelfand numbers tending to zero force compactness, provided the target space is complete.
Compactness is measured by the Gelfand numbers: for a complete target
space, S is a compact operator if and only if cₙ(S) → 0.
Hilbert spaces: every s-number sequence #
On Hilbert spaces, every compact operator is approximable. The singular
value decomposition IsCompactOperator.schmidtRepresentation produces singular values σₙ → 0
which, by Eckart–Young (svd_sigma_eq_approx), equal the approximation numbers
aₙ(S); hence aₙ(S) → 0.
On Hilbert spaces, approximable and compact operators coincide.
On Hilbert spaces every s-number sequence detects compactness. All
s-number sequences agree with the approximation numbers there
(SNumbers.allSNumbers_eq_on_HilbertSpace), and aₙ(S) → 0 is equivalent to
compactness by isApproximable_iff_isCompactOperator.
This fails on general Banach spaces for s = aₙ (Enflo), which is why the
Gelfand and Kolmogorov criteria in SNumbers are the ones stated there.
Hilbert-space case: compactness is equivalent to bₙ(S) → 0. On a general
Banach space only the forward implication holds.
Hilbert-space case: compactness is equivalent to hₙ(S) → 0, even though
the Hilbert numbers are the smallest s-number sequence.