Kolmogorov Numbers in Operator Approximation

• Kolmogorov numbers are a sequence of s-numbers that quantify the worst-case error of approximating an operator’s image via low-dimensional subspaces.
• They satisfy Pietsch’s axioms, ensuring monotonicity, multiplicativity, and normalization, and coincide with singular values in Hilbert spaces.
• They provide operational thresholds in communication channels, marking noise levels at which the channel’s effective dimensionality drops.

Searching arXiv for recent and foundational papers on Kolmogorov numbers to ground the article in published work. Kolmogorov numbers are an $s$-number sequence associated with a bounded linear operator $T:X\to Y$. They quantify the smallest worst-case error obtained by approximating the image of the unit ball $T(B_X)$ by subspaces of $Y$ of dimension $<n$. For compact operators they form a nonincreasing sequence tending to zero; in the Hilbert-space case they coincide with the singular values; and, in a communication-theoretic formulation, they are exactly the jump points of the degrees-of-freedom function $N(\epsilon)$ of a linear channel (Somaraju et al., 2011).

1. Definition and geometric content

Let $X$ and $Y$ be Banach spaces and let $T\colon X\to Y$ be bounded. If $Q_S^Y\colon Y\to Y/S$ denotes the quotient map onto the quotient by a closed subspace $T:X\to Y$0, then the $T:X\to Y$1th Kolmogorov number is

$T:X\to Y$2

Equivalently, $T:X\to Y$3 is the worst-case distance of $T:X\to Y$4 to an $T:X\to Y$5-dimensional subspace of $T:X\to Y$6. This makes explicit that Kolmogorov numbers are codomain-side approximation quantities: one fixes a low-dimensional subspace of $T:X\to Y$7 and measures how well the whole image $T:X\to Y$8 can be represented inside it (Somaraju et al., 2011).

A closely related finite-dimensional approximation formula is

$T:X\to Y$9

This version makes clear that the problem can be read as selecting $T(B_X)$0 trial vectors in the target space and minimizing the maximal residual over the unit ball of the domain (Somaraju et al., 2011).

The same basic definition extends to quasi-Banach spaces: $T(B_X)$1 In that setting, too, the quantity measures approximation of $T(B_X)$2 by finite-dimensional subspaces of the target (Gerhold, 5 Aug 2025).

2. Position within the theory of $T(B_X)$3-numbers

Kolmogorov numbers are one of the classical $T(B_X)$4-numbers in the sense of Pietsch. In the Banach-space setting they satisfy Pietsch’s axioms SN1–SN5 and therefore define a valid $T(B_X)$5-number sequence. In particular,

$T(B_X)$6

they obey the ideal property

$T(B_X)$7

they are normalized by $T(B_X)$8 on any Banach space, and they vanish once the rank drops below the index: $T(B_X)$9 These properties place them in the same structural framework as approximation, Gelfand, Bernstein, and Hilbert numbers [(Somaraju et al., 2011); (Ullrich, 2024)].

In quasi-Banach spaces, the structural picture persists with the expected modifications. One has monotonicity,

$Y$0

quasi-additivity,

$Y$1

multiplicativity,

$Y$2

the rank property, normalization for identities, and the ideal property

$Y$3

A central characterization is

$Y$4

which makes Kolmogorov numbers a compactness scale as well as an approximation scale (Gerhold, 5 Aug 2025).

A recent comparison theorem gives a sharp product-type bound between small and large $Y$5-numbers: $Y$6 Here $Y$7 are Gelfand numbers and $Y$8 are Hilbert numbers. This shows that Kolmogorov numbers cannot decay, on average, much faster than the Hilbert numbers up to the optimal factor $Y$9 (Ullrich, 2024).

3. Hilbert-space specialization and relations to neighboring quantities

When $<n$0 and $<n$1 are Hilbert spaces and $<n$2 is compact, the $<n$3-number theory collapses to the singular-value theory. In that case,

$<n$4

the $<n$5th singular value of $<n$6. This is one of the main reasons Kolmogorov numbers are viewed as the Banach-space analogue of singular values (Somaraju et al., 2011).

Several neighboring quantities recur throughout the literature.

Quantity	Notation	Relation stated in the sources
Approximation numbers	$<n$7	$<n$8
Gelfand numbers of the adjoint	$<n$9	$N(\epsilon)$0
Hilbert numbers	$N(\epsilon)$1	$N(\epsilon)$2
Entropy numbers	$N(\epsilon)$3	For compact Hilbert-to-Banach maps, decay of $N(\epsilon)$4 and $N(\epsilon)$5 is equivalent in several regimes
Interpolation widths	$N(\epsilon)$6	$N(\epsilon)$7

These comparisons clarify a common source of confusion: Kolmogorov numbers are not, in general, the same as approximation numbers, entropy numbers, or interpolation widths, even though they are often comparable (Steinwart, 2016, Ullrich, 2024).

For compact operators $N(\epsilon)$8 from a Hilbert space into a Banach space, Steinwart proves that for $N(\epsilon)$9,

$X$0

and, if $X$1 has the metric-extension property, analogous equivalences hold for $X$2 as well (Steinwart, 2016). In the setting of RKHS embeddings into $X$3, the same paper also exhibits a genuine half-power separation between Kolmogorov widths and interpolation widths: if $X$4 with $X$5, then

$X$6

The same $X$7 gap is attained for multidimensional Sobolev embeddings $X$8 (Steinwart, 2016).

A separate terminological issue appears in a recent Lie-group paper, where the covering number $X$9 is described as an “entropy Kolmogorov number,” while the standard Kolmogorov numbers remain the quantities $Y$0. In that setting the paper states

$Y$1

so the distinction between covering, entropy, and Kolmogorov numbers is preserved even when the terminology is broadened (Avetisyan et al., 2 Feb 2026).

4. Communication channels and the degrees-of-freedom function

A linear communication channel may be modeled by a compact linear operator $Y$2, where $Y$3 is a transmitter power constraint and $Y$4 is a receiver noise threshold below which signals cannot be distinguished. In that setting the number of degrees of freedom at level $Y$5 is defined by

$Y$6

Operationally, $Y$7 is the number of linearly independent signals that may be communicated through the channel at noise level $Y$8 (Somaraju et al., 2011).

The function $Y$9 has the basic properties expected of a degrees-of-freedom count. One has $T\colon X\to Y$0 for all $T\colon X\to Y$1, the map is nonincreasing, and it has only finitely many jumps in any finite $T\colon X\to Y$2-interval. Unless $T\colon X\to Y$3, $T\colon X\to Y$4 as $T\colon X\to Y$5 (Somaraju et al., 2011).

If $T\colon X\to Y$6 denotes the $T\colon X\to Y$7th jump point of $T\colon X\to Y$8, characterized by

$T\colon X\to Y$9

then the central theorem is

$Q_S^Y\colon Y\to Y/S$0

Thus Kolmogorov numbers are exactly the discontinuity thresholds of the degrees-of-freedom function. This gives an operator-theoretic quantity an immediate communication-theoretic interpretation: the $Q_S^Y\colon Y\to Y/S$1th Kolmogorov number is the noise level at which the effective dimensionality of the channel drops from at least $Q_S^Y\colon Y\to Y/S$2 to at most $Q_S^Y\colon Y\to Y/S$3 (Somaraju et al., 2011).

5. Asymptotic theories for embeddings and finite-dimensional models

A major part of the literature studies asymptotics of $Q_S^Y\colon Y\to Y/S$4 for concrete embeddings. For weighted Sobolev-type embeddings of Besov and Triebel–Lizorkin spaces with polynomial weights,

$Q_S^Y\colon Y\to Y/S$5

and analogous $Q_S^Y\colon Y\to Y/S$6-space variants, the non-limiting case is governed by

$Q_S^Y\colon Y\to Y/S$7

The sharp asymptotics are of the form

$Q_S^Y\colon Y\to Y/S$8

with six parameter regimes determined by $Q_S^Y\colon Y\to Y/S$9, and the same rates remain valid for the corresponding Besov/Triebel–Lizorkin combinations under the stated quasi-normability restrictions [(Zhang et al., 2011); (Zhang et al., 2011)].

The proofs in these weighted function-space problems proceed by wavelet discretization, reduction to sequence-space embeddings, and use of sharp finite-dimensional $T:X\to Y$00 estimates. One representative finite-dimensional identity is

$T:X\to Y$01

while other parameter ranges exhibit piecewise asymptotics involving logarithmic factors, $T:X\to Y$02-decay, or interpolation between two regimes (Gerhold, 5 Aug 2025). These finite-dimensional estimates are also used explicitly in the weighted Sobolev papers through dyadic decomposition and operator-ideal arguments (Zhang et al., 2011).

For Schatten-class embeddings

$T:X\to Y$03

Prochno and Strzelecki obtain asymptotically sharp two-sided bounds, up to constants depending only on $T:X\to Y$04, across several parameter regions. For example, in the “upper-triangle” regime $T:X\to Y$05,

$T:X\to Y$06

In the quasi-Banach range $T:X\to Y$07, the same paper states

$T:X\to Y$08

The analysis relies on duality with Gelfand numbers, volume/Dvoretzky arguments, interpolation, and explicit construction of quotients of $T:X\to Y$09 of controlled norm (Prochno et al., 2021).

Another strand concerns embeddings of Besov spaces of dominating mixed smoothness into $T:X\to Y$10. The abstract of "Kolmogorov Numbers of Embeddings of Besov Spaces of Dominating Mixed Smoothness into $T:X\to Y$11" announces two-sided sharp estimates for

$T:X\to Y$12

placing mixed-smoothness problems within the same general width-theoretic framework (Nguyen, 2014).

6. Numerical computation and further analytic settings

For computation, a practical scheme is available when the domain $T:X\to Y$13 has a Schauder basis $T:X\to Y$14. If

$T:X\to Y$15

and $T:X\to Y$16, then for each fixed $T:X\to Y$17,

$T:X\to Y$18

once $T:X\to Y$19 is large enough that the $T:X\to Y$20th Kolmogorov number of the truncation exists. This leads to a finite-dimensional approximation procedure: choose $T:X\to Y$21, assemble the matrix of $T:X\to Y$22, compute $T:X\to Y$23, and increase $T:X\to Y$24 until the values stabilize. In the Hilbert-space case this reduces to extracting the $T:X\to Y$25th singular value of the finite matrix, or equivalently diagonalizing the associated Gram matrix (Somaraju et al., 2011).

The same approximation-theoretic viewpoint now appears in newer geometric settings. For a compact Lie group $T:X\to Y$26, a left-invariant positive symmetric trace-class integral kernel $T:X\to Y$27 induces an RKHS $T:X\to Y$28, and one studies the embedding

$T:X\to Y$29

In that context, the paper on compact Lie groups defines covering numbers $T:X\to Y$30, entropy numbers $T:X\to Y$31, and Kolmogorov numbers $T:X\to Y$32, and proves two-sided asymptotic estimates for $T:X\to Y$33 from spectral assumptions on the group Fourier symbol of the kernel operator. Under a trace-decay hypothesis $T:X\to Y$34 with $T:X\to Y$35, the paper derives power-law decay of $T:X\to Y$36 up to logarithmic factors; under a determinant-decay hypothesis, it derives logarithmic asymptotics of the form $T:X\to Y$37 in the respective regime (Avetisyan et al., 2 Feb 2026).

Taken together, these results show that Kolmogorov numbers serve simultaneously as abstract $T:X\to Y$38-numbers, as singular-value analogues on Banach and quasi-Banach spaces, as operational thresholds in communication channels, and as sharp asymptotic invariants for embeddings in function spaces, matrix ideals, RKHS theory, and harmonic analysis on compact groups [(Somaraju et al., 2011); (Steinwart, 2016); (Prochno et al., 2021); (Avetisyan et al., 2 Feb 2026)].