Ergodic Hankel Operators: Dynamics & Spectra

• Ergodic Hankel operators are defined in two frameworks: as maps between Köthe/power series spaces encoding shift dynamics, and as self-adjoint integral operators on L2(0,∞) analyzed via spectral methods.
• In the Köthe-space setting, criteria for continuity and compactness connect Hankel columns with forward/backward shifts and Toeplitz operators, ensuring mean ergodicity in Cesàro averages.
• The spectral framework employs ergodic covariance, Integrated Density of States, and random models to reveal phenomena such as Lifshitz tails and Anderson localization.

Ergodic Hankel operators arise in two distinct but related senses in recent research. In one sense, Hankel operators are studied between Köthe and power series spaces, where ergodic phenomena enter through the backward and forward shift operators and through the Cesàro behavior of the columns of a Hankel matrix (Doğan, 2024). In a second, more directly spectral sense, ergodic Hankel operators are introduced as ergodic families of bounded self-adjoint Hankel integral operators on $L^2(0,\infty)$, realized after transplantation as covariant operators on $L^2(\mathbb{R})$ and analyzed via the Integrated Density of States (IDS), Floquet theory, and random-operator methods (Pastur et al., 29 Sep 2025). The term therefore does not refer to a single universal formalism; rather, it currently designates both a dynamical viewpoint on Hankel-generated shift behavior and a spectral theory of ergodic operator families.

1. Terminological scope and basic operator models

A Hankel operator is characterized by dependence on the sum of indices or variables. In the sequence-space setting, a Hankel matrix $(a_{m,n})$ is one whose entries are constant along skew diagonals, and for a sequence $\theta=(\theta_n)_{n\in\mathbb{N}_0}$ the associated infinite Hankel matrix is

$H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$

When defined between Köthe spaces, its action on the canonical basis is

$$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$

and for $x=\sum_{n=1}^\infty x_n e_n$ one sets

$H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$

provided the series is well defined (Doğan, 2024).

In the half-line integral setting, a Hankel operator $H$ on $L^2(0,\infty)$ is realized by

$L^2(\mathbb{R})$0

with kernel $L^2(\mathbb{R})$1. If $L^2(\mathbb{R})$2 is real-valued, the operator is self-adjoint. A sufficient boundedness condition used throughout is

$L^2(\mathbb{R})$3

which, via the Carleman estimate,

$L^2(\mathbb{R})$4

implies $L^2(\mathbb{R})$5 (Pastur et al., 29 Sep 2025).

A central structural device in the half-line theory is the unitary transplantation

$L^2(\mathbb{R})$6

from $L^2(\mathbb{R})$7 to $L^2(\mathbb{R})$8. If $L^2(\mathbb{R})$9, then

$(a_{m,n})$0

For kernels of the form $(a_{m,n})$1, this becomes

$(a_{m,n})$2

which makes the covariance structure of ergodic families explicit after passing to logarithmic coordinates (Pastur et al., 29 Sep 2025).

This bifurcation of models is crucial. In the Köthe-space literature, ergodicity is not attributed to the powers $(a_{m,n})$3 themselves. In the half-line theory, by contrast, ergodicity is built into a covariant family $(a_{m,n})$4 indexed by an ergodic probability space (Doğan, 2024, Pastur et al., 29 Sep 2025).

2. Hankel operators between Köthe and power series spaces

The sequence-space framework begins with Köthe spaces

$(a_{m,n})$5

where $(a_{m,n})$6 is a Köthe matrix. Every Köthe space is a Fréchet space, and its dual is

$(a_{m,n})$7

(Doğan, 2024).

The principal concrete subclasses are the power series spaces associated with a non-negative increasing sequence $(a_{m,n})$8 satisfying $(a_{m,n})$9:

$\theta=(\theta_n)_{n\in\mathbb{N}_0}$0

$\theta=(\theta_n)_{n\in\mathbb{N}_0}$1

These are Köthe spaces, and every power series space is Montel. The examples

$\theta=(\theta_n)_{n\in\mathbb{N}_0}$2

place the theory in a standard holomorphic-function context (Doğan, 2024).

For Hankel operators between such spaces, the basic continuity criterion is expressed on the canonical basis. The operator $\theta=(\theta_n)_{n\in\mathbb{N}_0}$3 is well defined and continuous if and only if

1. $\theta=(\theta_n)_{n\in\mathbb{N}_0}$4 for every $\theta=(\theta_n)_{n\in\mathbb{N}_0}$5, and
2. for every $\theta=(\theta_n)_{n\in\mathbb{N}_0}$6 there exists $\theta=(\theta_n)_{n\in\mathbb{N}_0}$7 such that

$\theta=(\theta_n)_{n\in\mathbb{N}_0}$8

A necessary consequence is that if $\theta=(\theta_n)_{n\in\mathbb{N}_0}$9 is continuous, then $H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$0 and also $H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$1 (Doğan, 2024).

The paper develops several sufficient conditions for continuity and compactness, especially for mappings between power series spaces. Among the principal statements are the following.

• For every $H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$2, the Hankel operator

$H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$3

is continuous and compact (Doğan, 2024).

• If $H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$4 for all $H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$5 and some $H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$6, then for every $H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$7,

$H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$8

is well defined, continuous, and compact (Doğan, 2024).

• If $H_\theta=[\theta_{i+j}]_{i,j\ge 0}.$9 is stable and $$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$0 for all $$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$1, then for every $$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$2,

$$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$3

is well defined, continuous, and compact (Doğan, 2024).

• If $$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$4 is a nuclear finite-type power series space, then for every $$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$5,

$$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$6

is well defined, continuous, and compact (Doğan, 2024).

These results show that, in the Köthe-space setting, compactness is pervasive under natural growth assumptions. This suggests that the ergodic questions relevant there are less about long-time iteration of a fixed Hankel operator and more about how Hankel structure encodes the dynamics of related operators, especially shifts.

3. Shift dynamics and the first ergodic interpretation

The ergodic content of the Köthe-space theory is mediated by the backward and forward shift operators on $$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$7, $$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$8:

$$H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j,$$9

$x=\sum_{n=1}^\infty x_n e_n$0

These operators are well defined and continuous when $x=\sum_{n=1}^\infty x_n e_n$1 is weakly-stable, meaning

$x=\sum_{n=1}^\infty x_n e_n$2

and the paper works under the stronger assumption that $x=\sum_{n=1}^\infty x_n e_n$3 is stable (Doğan, 2024).

A Toeplitz operator associated with $x=\sum_{n=1}^\infty x_n e_n$4 is also introduced:

$x=\sum_{n=1}^\infty x_n e_n$5

The decisive identities are

$x=\sum_{n=1}^\infty x_n e_n$6

They identify the iterates of the forward shift with Toeplitz columns and the iterates of the backward shift with Hankel columns (Doğan, 2024).

For a continuous operator $x=\sum_{n=1}^\infty x_n e_n$7 on a Fréchet space $x=\sum_{n=1}^\infty x_n e_n$8, the $x=\sum_{n=1}^\infty x_n e_n$9-th Cesàro mean is

$H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$0

The operator is mean ergodic if $H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$1 exists in $H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$2 for every $H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$3, and it is Cesàro bounded if $H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$4 is equicontinuous in $H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$5. On a Montel Fréchet space, Kalmes–Santacreu’s criterion gives that $H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$6 is mean ergodic if and only if it is Cesàro bounded and

$H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$7

for every $H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$8 (Doğan, 2024).

Using the identities above and continuity of the Toeplitz and Hankel actions, the paper proves

$H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n$9

for all $H$0, $H$1. Concretely,

$H$2

Accordingly, the forward and backward shifts on $H$3, for stable $H$4, are mean ergodic and Cesàro bounded (Doğan, 2024).

This gives the first modern meaning of “ergodic Hankel operators”: not that $H$5 itself is shown to be mean ergodic under iteration, but that Hankel columns realize the iterates of an ergodic shift. A common misconception is therefore to equate the result with a theorem about the Cesàro means of $H$6. The paper explicitly does not analyze mean ergodicity or Cesàro boundedness of the powers $H$7 themselves (Doğan, 2024).

4. Ergodic families on the half-line

A distinct theory is developed for bounded self-adjoint Hankel operators on $H$8 that form ergodic families under dilations (Pastur et al., 29 Sep 2025). In logarithmic coordinates, these become covariant families on $H$9.

Let $L^2(0,\infty)$0 be a probability space with an ergodic group $L^2(0,\infty)$1, where $L^2(0,\infty)$2 in the continuous case or $L^2(0,\infty)$3 in the discrete case. Let

$L^2(0,\infty)$4

for a fixed period $L^2(0,\infty)$5. A measurable family of bounded operators $L^2(0,\infty)$6 on $L^2(0,\infty)$7 is ergodic with period $L^2(0,\infty)$8 if

$L^2(0,\infty)$9

In $L^2(\mathbb{R})$00 this is equivalent to

$L^2(\mathbb{R})$01

where

$L^2(\mathbb{R})$02

Thus the natural ergodic symmetry for half-line Hankel operators is multiplicative in the original variable and translational after logarithmic transplantation (Pastur et al., 29 Sep 2025).

If one writes

$L^2(\mathbb{R})$03

then ergodicity is equivalent to

$L^2(\mathbb{R})$04

Under the standing bound $L^2(\mathbb{R})$05 almost surely, one has

$L^2(\mathbb{R})$06

almost surely (Pastur et al., 29 Sep 2025).

The positive case has an especially rigid structure. A bounded Hankel operator is positive if and only if

$L^2(\mathbb{R})$07

with $L^2(\mathbb{R})$08 and the Carleson condition

$L^2(\mathbb{R})$09

For positive ergodic Hankel operators there exists an ergodic family of Borel measures $L^2(\mathbb{R})$10 on $L^2(\mathbb{R})$11 such that

$L^2(\mathbb{R})$12

and

$L^2(\mathbb{R})$13

with

$L^2(\mathbb{R})$14

The covariance condition becomes

$L^2(\mathbb{R})$15

for Borel $L^2(\mathbb{R})$16, together with the uniform local boundedness estimate

$L^2(\mathbb{R})$17

uniformly in $L^2(\mathbb{R})$18 and $L^2(\mathbb{R})$19 (Pastur et al., 29 Sep 2025).

This framework produces a spectral theory analogous in several respects to that of ergodic Schrödinger operators, but with genuinely Hankel-specific geometry arising from the $L^2(\mathbb{R})$20 kernel dependence and the dilation covariance.

5. Integrated Density of States and structural spectral results

The Integrated Density of States is the central invariant of the ergodic half-line theory. For a Borel set $L^2(\mathbb{R})$21 separated from $L^2(\mathbb{R})$22, the IDS measure $L^2(\mathbb{R})$23 is defined by

$L^2(\mathbb{R})$24

in the continuous case, and by

$L^2(\mathbb{R})$25

in the discrete case, with $L^2(\mathbb{R})$26 by convention (Pastur et al., 29 Sep 2025).

Finite-volume approximations yield self-averaging. Almost surely,

$L^2(\mathbb{R})$27

where $L^2(\mathbb{R})$28. Equivalently, for continuous $L^2(\mathbb{R})$29 compactly supported in $L^2(\mathbb{R})$30,

$L^2(\mathbb{R})$31

The support of $L^2(\mathbb{R})$32 coincides with the almost-sure spectrum of $L^2(\mathbb{R})$33 (Pastur et al., 29 Sep 2025).

Two Szegő-type formulations are established. First,

$L^2(\mathbb{R})$34

Second, in the positive case, if $L^2(\mathbb{R})$35 is obtained by restricting $L^2(\mathbb{R})$36 to $L^2(\mathbb{R})$37, then

$L^2(\mathbb{R})$38

for continuous $L^2(\mathbb{R})$39 supported away from $L^2(\mathbb{R})$40 (Pastur et al., 29 Sep 2025).

Several general properties follow.

• The spectrum and its multiplicity are non-random (Pastur et al., 29 Sep 2025).
• The second moment of the IDS is finite in general, and the first moment is finite if $L^2(\mathbb{R})$41 is positive (Pastur et al., 29 Sep 2025).
• If $L^2(\mathbb{R})$42 is positive, then $L^2(\mathbb{R})$43 has no atoms (Pastur et al., 29 Sep 2025).

A particularly notable theorem identifies the total mass of the positive IDS. If $L^2(\mathbb{R})$44 is positive and bounded with associated $L^2(\mathbb{R})$45, and

$L^2(\mathbb{R})$46

then

$L^2(\mathbb{R})$47

In the terminology of the paper, $L^2(\mathbb{R})$48 is the mean number of spectral bands per unit period, counting flat and non-flat bands but ignoring $L^2(\mathbb{R})$49; it equals the mean number of atoms of $L^2(\mathbb{R})$50 per period (Pastur et al., 29 Sep 2025).

The Carleman operator $L^2(\mathbb{R})$51 provides an explicit benchmark. In that case, the IDS satisfies

$L^2(\mathbb{R})$52

with density

$L^2(\mathbb{R})$53

In particular,

$L^2(\mathbb{R})$54

This explicit asymptotic makes precise the logarithmic singularity at zero in the model case (Pastur et al., 29 Sep 2025).

6. Periodic and random ergodic Hankel operators

The periodic theory begins with $L^2(\mathbb{R})$55-periodic kernels of the form

$L^2(\mathbb{R})$56

where $L^2(\mathbb{R})$57 is $L^2(\mathbb{R})$58-periodic. In $L^2(\mathbb{R})$59, the corresponding operator commutes with $L^2(\mathbb{R})$60, and this admits a Floquet–Bloch decomposition

$L^2(\mathbb{R})$61

For smooth periodic Hankel operators satisfying

$L^2(\mathbb{R})$62

the fibers $L^2(\mathbb{R})$63 are trace class and have matrix entries

$L^2(\mathbb{R})$64

where $L^2(\mathbb{R})$65 (Pastur et al., 29 Sep 2025).

The nonzero spectrum is organized by analytic band functions $L^2(\mathbb{R})$66. The IDS in the periodic case has no singular continuous part. More precisely,

$L^2(\mathbb{R})$67

where each $L^2(\mathbb{R})$68 is a purely absolutely continuous probability measure supported on a closed interval $L^2(\mathbb{R})$69, and

$L^2(\mathbb{R})$70

where $L^2(\mathbb{R})$71 are flat band energies. For positive periodic Hankel operators, there are no flat bands, so $L^2(\mathbb{R})$72 (Pastur et al., 29 Sep 2025).

Two explicit periodic examples illustrate the range of behaviors.

• If $L^2(\mathbb{R})$73, then $L^2(\mathbb{R})$74 is rank one and its nonzero eigenvalue is

$L^2(\mathbb{R})$75

The resulting single band is absolutely continuous and has square-root edge singularities (Pastur et al., 29 Sep 2025).

• If

$L^2(\mathbb{R})$76

then $L^2(\mathbb{R})$77 is rank two with constant eigenvalues $L^2(\mathbb{R})$78, and the IDS is pure point with two atoms at $L^2(\mathbb{R})$79 (Pastur et al., 29 Sep 2025).

The random Kronig–Penney–Hankel (rKPH) model supplies the random counterpart. It is defined on $L^2(\mathbb{R})$80 by

$L^2(\mathbb{R})$81

where $L^2(\mathbb{R})$82, $L^2(\mathbb{R})$83, and $L^2(\mathbb{R})$84 are i.i.d. random variables with compact support

$L^2(\mathbb{R})$85

Its associated ergodic measure is

$L^2(\mathbb{R})$86

(Pastur et al., 29 Sep 2025).

Let

$L^2(\mathbb{R})$87

and let $L^2(\mathbb{R})$88 be the range extrema of $L^2(\mathbb{R})$89. Then the almost-sure spectrum is

$L^2(\mathbb{R})$90

so in particular the endpoints $L^2(\mathbb{R})$91 and $L^2(\mathbb{R})$92 belong to the spectrum (Pastur et al., 29 Sep 2025).

For this model, the paper proves the analogues of the standard one-dimensional random-operator cornerstones.

• Total mass of the IDS:

$L^2(\mathbb{R})$93

because there is one atom of $L^2(\mathbb{R})$94 per cell almost surely (Pastur et al., 29 Sep 2025).

• Lifshitz tails: if $L^2(\mathbb{R})$95 is not supported at a point and satisfies

$L^2(\mathbb{R})$96

for small $L^2(\mathbb{R})$97, then

$L^2(\mathbb{R})$98

$L^2(\mathbb{R})$99

The exponent is the one-dimensional Lifshitz exponent $(a_{m,n})$00 (Pastur et al., 29 Sep 2025).

• Wegner bound: if $(a_{m,n})$01 has bounded density $(a_{m,n})$02, then $(a_{m,n})$03 is absolutely continuous with bounded density and for every interval $(a_{m,n})$04,

$(a_{m,n})$05

In finite volume, this corresponds to

$(a_{m,n})$06

(Pastur et al., 29 Sep 2025).

• Anderson localization: if $(a_{m,n})$07 is uniformly Hölder continuous of order $(a_{m,n})$08 on $(a_{m,n})$09, then there exists $(a_{m,n})$10 such that for $(a_{m,n})$11, the operator $(a_{m,n})$12 has pure point spectrum almost surely, with exponentially decaying eigenfunctions (Pastur et al., 29 Sep 2025).

These results establish an operator-theoretic analogue of periodic and random Schrödinger theory in a genuinely Hankel setting.

7. Conceptual distinctions, limitations, and open directions

The recent literature makes clear that “ergodic Hankel operators” has two non-equivalent uses. The sequence-space literature analyzes Hankel operators as maps between Köthe or power series spaces and derives ergodic conclusions for the backward and forward shifts through the identities

$(a_{m,n})$13

Its main ergodic conclusion is the vanishing of Cesàro averages of shift iterates, equivalently the Cesàro averages of Hankel columns, not the mean ergodicity of the operator powers $(a_{m,n})$14 (Doğan, 2024).

By contrast, the half-line theory introduces ergodic families of Hankel operators in the same structural sense in which one speaks of ergodic Schrödinger operators: a probability space, a covariant group action, self-averaging spectral data, an IDS, and periodic/random models (Pastur et al., 29 Sep 2025). A plausible implication is that the latter framework will likely become the standard meaning of the term in spectral theory, while the former remains indispensable for locally convex and function-space operator theory.

Each framework also has explicit limitations. In the Köthe-space setting, the paper does not provide criteria for mean ergodicity or Cesàro boundedness of $(a_{m,n})$15 itself under iteration, and the sufficient continuity and compactness conditions depend on stability assumptions and inequalities relating Köthe weights to exponential weights (Doğan, 2024). In the half-line setting, the theory is currently formulated for bounded operators, with positivity playing a particularly strong role through the Laplace-transform representation, the Carleson condition, and continuity of the IDS (Pastur et al., 29 Sep 2025).

The open problems stated in the spectral theory of ergodic Hankel operators are correspondingly broad. They include quasi- or almost-periodic Hankel operators, unbounded ergodic Hankel operators, detailed IDS asymptotics at zero, coexistence and interaction of flat and non-flat bands, and the development of Hankel analogues of transfer matrices, Lyapunov exponents, and $(a_{m,n})$16-functions (Pastur et al., 29 Sep 2025). From the perspective of the Köthe-space theory, a natural unresolved direction is a direct study of ergodicity for the powers or Cesàro means of Hankel operators themselves rather than only for the shifts they encode (Doğan, 2024).

Taken together, these two strands show that ergodic Hankel operators now occupy a dual position in operator theory: as structured maps between Köthe-type spaces whose columns exhibit ergodic averaging through shift dynamics, and as a new class of ergodic self-adjoint Hankel families with a developed IDS theory, periodic band structure, and random localization phenomena (Doğan, 2024, Pastur et al., 29 Sep 2025).