Periodic Hankel Operators

• Periodic Hankel Operators are Hankel operators with an added periodic structure, defined via multiplicative, Fourier, or symbol-based settings.
• The half-line model uses dilation-periodicity and Floquet–Bloch theory to reveal band spectra with both dispersive (non-flat) and constant (flat) bands.
• On the circle and group settings, periodic Hankel operators are characterized by Fourier data and symbol jump conditions, enabling detailed spectral and index analysis.

Searching arXiv for recent and foundational papers on periodic Hankel operators across half-line, circle/Hardy-space, and spectral-density settings. {"query":"periodic Hankel operators Floquet Bloch band spectrum site:arxiv.org"} {"query":"arXiv (Pastur et al., 29 Sep 2025) Ergodic Hankel operators periodic Hankel operators"} Periodic Hankel operators are Hankel operators endowed with an additional periodic structure, but the phrase is used in several closely related senses. In the half-line theory, the periodicity is multiplicative: the kernel has the form

$$(Hf)(t)=\int_0^\infty h(t+s)f(s)\,ds,\qquad h(t)=\frac{P(\log t)}{t},$$

with $P$ periodic in $\log t$, so $H$ commutes with a discrete dilation. In the circle and Hardy-space literature, the same phrase refers to Hankel operators on $\mathbb T$, where periodicity is built into the underlying domain and the operator is generated by Fourier data or Hardy symbols. A broader harmonic-analytic generalization places these constructions on compact connected Abelian groups with ordered duals (Pastur et al., 29 Sep 2025, Gerard et al., 2014, Mirotin, 2022, Mirotin et al., 2016).

1. Terminology and operator models

The common structural feature is the Hankel dependence on a sum of variables or indices. On $L^2(\mathbb R_+)$, the kernel depends on $t+s$; on $\ell^2(\mathbb Z_+)$, the matrix entries depend on $j+k$; on Hardy space $H^2(\mathbb T)$, the operator is realized by projecting a product onto anti-analytic modes. In the half-line framework developed for ergodic and periodic Hankel operators, bounded self-adjoint Hankel operators are realized as

$P$0

with boundedness ensured, for example, by $P$1, and self-adjointness equivalent to $P$2 being real-valued (Pastur et al., 29 Sep 2025).

A distinct but standard periodic model is the circle Hardy-space operator

$P$3

which is antilinear, or its linear companion

$P$4

acting from $P$5 to $P$6. In matrix form, these correspond to classical Hankel matrices constant on anti-diagonals (Gerard et al., 2014, Sołtysiak, 2022). On $P$7, the symbol-generated Hankel matrix is

$P$8

unitarily equivalent to the Hardy-space operator

$P$9

with self-adjointness characterized by

$\log t$0

This circle model is explicitly treated as a periodic Hankel setting in the spectral-density theory for piecewise continuous symbols (Fedele, 2019).

A plausible implication is that “periodic Hankel operator” is not a single universal definition but a family of constructions unified by Hankel symmetry together with periodicity of the ambient variable, the logarithmic variable, or the generating symbol.

2. Dilation-periodic operators on the half-line

The modern half-line theory identifies periodicity with discrete dilation covariance. If

$\log t$1

then the corresponding Hankel operator commutes with

$\log t$2

that is,

$\log t$3

After the unitary logarithmic change of variables

$\log t$4

the transformed operator $\log t$5 acts on $\log t$6 and commutes with translation by $\log t$7: $\log t$8 This makes Floquet–Bloch theory available in direct analogy with periodic Schrödinger operators (Pastur et al., 29 Sep 2025).

The smooth periodic class is defined by the Fourier coefficients

$\log t$9

with the summability condition

$H$0

Under this assumption, the fiber operators in the Floquet decomposition are trace class (Pastur et al., 29 Sep 2025).

The fiber Hilbert space is $H$1, and for a bounded self-adjoint operator commuting with $H$2 one has

$H$3

with

$H$4

Applied to a periodic Hankel operator, this yields

$H$5

A concrete fiber formula is available: $H$6 or equivalently in the factorized Gamma-form recorded in the paper. This formula is central because it converts the dilation-periodic Hankel operator into a compact-operator family parametrized by quasi-momentum $H$7 (Pastur et al., 29 Sep 2025).

An earlier treatment of essentially the same class establishes the same Floquet–Bloch framework with period $H$8, writing $H$9 and fiber matrices through the Beta function. There the key observation is that logarithmic variables convert discrete dilations into translations, so Mellin analysis plays the role normally played by Fourier analysis in periodic operator theory (Pushnitski et al., 2023).

3. Band spectra, flat bands, and the integrated density of states

For smooth periodic Hankel operators on the half-line, the nonzero spectrum is described by analytic band functions. There exists a finite or countable list of nonvanishing real-analytic functions $\mathbb T$0, called band functions, representing all nonzero fiber eigenvalues with multiplicity encoded by repetition. The list splits into non-flat and flat bands. Non-flat bands have multiplicity one and satisfy

$\mathbb T$1

while flat bands are constant. Distinct non-flat bands have disjoint interiors, and no non-flat band intersects the negative of another (Pastur et al., 29 Sep 2025).

The integrated density of states for smooth periodic Hankel operators admits a complete decomposition. If $\mathbb T$2 denotes the IDS measure, then

$\mathbb T$3

The absolutely continuous part has the form

$\mathbb T$4

where each $\mathbb T$5 is a purely absolutely continuous probability measure supported on a closed bounded interval $\mathbb T$6 separated away from $\mathbb T$7. The pure point part has the form

$\mathbb T$8

with $\mathbb T$9 if the family is infinite; the points $L^2(\mathbb R_+)$0 are eigenvalues of infinite multiplicity and are interpreted as flat spectral bands. If the operator is positive, then $L^2(\mathbb R_+)$1, so the IDS is purely absolutely continuous (Pastur et al., 29 Sep 2025).

The support of the IDS equals the deterministic spectrum: $L^2(\mathbb R_+)$2 Thus periodic half-line Hankel operators have a deterministic band spectrum consisting of absolutely continuous bands, possibly together with flat bands at isolated nonzero eigenvalues, and with $L^2(\mathbb R_+)$3 as the only possible accumulation point if infinitely many bands occur (Pastur et al., 29 Sep 2025).

The band picture was already developed in detail in the earlier band-spectrum analysis. There the absolutely continuous spectrum has multiplicity $L^2(\mathbb R_+)$4, the singular continuous spectrum is absent, and nonzero eigenvalues have infinite multiplicity. A particularly distinctive feature is that flat bands may coexist with non-flat bands, unlike the standard one-dimensional periodic Schrödinger picture. The paper’s Mathieu–Hankel operator provides an explicit example: there exists $L^2(\mathbb R_+)$5 such that $L^2(\mathbb R_+)$6 has at least one flat band and at least one non-flat band (Pushnitski et al., 2023).

The periodic IDS also satisfies a gap-labelling statement: in a spectral gap,

$L^2(\mathbb R_+)$7

is constant and equals $L^2(\mathbb R_+)$8, where $L^2(\mathbb R_+)$9 is the number of bands above $t+s$0, counting both flat and non-flat bands (Pastur et al., 29 Sep 2025).

4. Periodic Hankel operators on the circle

On the circle, periodicity is intrinsic. The Hardy-space model takes

$t+s$1

and defines the compact Hankel operator

$t+s$2

with compactness equivalent to $t+s$3. The shifted operator

$t+s$4

is essential because the inverse spectral problem is controlled by the pair $t+s$5, not by $t+s$6 alone. The relation

$t+s$7

drives interlacing and multiplicity phenomena (Gerard et al., 2014).

For compact periodic Hankel operators on the circle, singular values with arbitrary multiplicities are classified by a nonlinear Fourier transform. The complete spectral data consist of the interlaced singular values of $t+s$8 and $t+s$9, together with one finite Blaschke product $\ell^2(\mathbb Z_+)$0 for each singular value $\ell^2(\mathbb Z_+)$1. The map

$\ell^2(\mathbb Z_+)$2

is bijective, and in finite rank its inverse is explicit. A singular value of multiplicity $\ell^2(\mathbb Z_+)$3 contributes a Blaschke product of degree $\ell^2(\mathbb Z_+)$4 (Gerard et al., 2014).

The same circle setting also supports asymptotic spectral analysis for noncompact symbol-defined periodic Hankel matrices. For $\ell^2(\mathbb Z_+)$5, the Hankel matrix

$\ell^2(\mathbb Z_+)$6

has truncations whose logarithmic spectral density depends only on the jump half-heights

$\ell^2(\mathbb Z_+)$7

In the non-self-adjoint case,

$\ell^2(\mathbb Z_+)$8

and in the self-adjoint case the positive and negative densities split according to conjugate jump pairs and the special points $\ell^2(\mathbb Z_+)$9. The result is universal with respect to the truncation scheme under the paper’s assumptions: square truncation, Abel–Poisson regularization, and other admissible Schur–Hadamard multipliers give the same asymptotic logarithmic density (Fedele, 2019).

A more geometric analysis of circle Hankel operators studies Schmidt subspaces

$j+k$0

for the linear Hardy-space Hankel operator $j+k$1. For the top singular value $j+k$2, the Schmidt subspace is exactly the kernel of a Toeplitz operator. More generally,

$j+k$3

and if $j+k$4, then it is nearly $j+k$5-invariant (Sołtysiak, 2022). This does not define periodicity, but it gives structural information for periodic circle symbols as a subclass.

5. Group-theoretic and symbol-theoretic extensions

The circle theory extends to compact connected Abelian groups $j+k$6 with linearly ordered dual $j+k$7. The Hardy spaces are defined by Fourier support in a positive cone $j+k$8: $j+k$9 and the classical Hankel operator is

$H^2(\mathbb T)$0

When $H^2(\mathbb T)$1, this recovers the usual periodic Hardy-space Hankel operator. In this framework, boundedness is characterized by $H^2(\mathbb T)$2-symbol extension or, equivalently, by $H^2(\mathbb T)$3; compactness is characterized by $H^2(\mathbb T)$4 when the dual has a least positive element; and bounded Hankel operators are not left Fredholm (Mirotin et al., 2016).

A different generalization introduces $H^2(\mathbb T)$5-Hankel operators on Hardy spaces over compact Abelian groups. They are defined by

$H^2(\mathbb T)$6

For $H^2(\mathbb T)$7, this becomes

$H^2(\mathbb T)$8

which is exactly the weighted Hankel anti-diagonal pattern. If $H^2(\mathbb T)$9 for all $P$00, every bounded $P$01-Hankel operator is of the form

$P$02

for a unitary translation $P$03 and a classical Hankel operator $P$04 (Mirotin, 2022).

For symbol-defined Hankel matrices on $P$05, the Fredholm and index theory is most naturally formulated inside the Toeplitz-plus-Hankel algebra $P$06. There, the local symbol of

$P$07

is a $P$08 matrix away from $P$09 and a scalar at $P$10, with the Hankel contribution entering through the jump differences $P$11. The main message for structured or periodic symbol-generated Hankel matrices is that Fredholmness is governed explicitly by local jump data of the symbol, and the index is a winding number of the resulting normalized symbol. The same paper also notes that a pure Hankel operator $P$12 alone can only be Fredholm in very exceptional circumstances within this algebraic setting, because the Toeplitz part $P$13 must be invertible (Roch et al., 2011).

6. Explicit examples, characteristic phenomena, and scope

The periodic half-line theory has several explicit examples that isolate genuinely Hankel features. If

$P$14

then the fibers have rank one and there is a single nonzero band

$P$15

so the IDS is absolutely continuous on $P$16; because the extrema are nondegenerate, the density has square-root singularities near the endpoints (Pastur et al., 29 Sep 2025).

A signed periodic measure gives the opposite extreme. For

$P$17

the fibers are rank two and their eigenvalues are $P$18, but an elliptic-function identity makes $P$19 independent of $P$20. Hence both bands are flat, the spectrum consists of

$P$21

and the IDS is pure point, supported at $P$22. This is the explicit example showing that, without positivity, pure point IDS can occur (Pastur et al., 29 Sep 2025).

The earlier band-spectrum paper gives further explicit models. For the Carleman operator, all bands are positive and non-flat, filling $P$23 with multiplicity $P$24. For the Mathieu–Hankel operator with

$P$25

one has all bands flat when $P$26, infinitely many positive and negative bands for $P$27, positivity of all fibers for $P$28, and existence of a parameter $P$29 for which flat and non-flat bands coexist (Pushnitski et al., 2023).

Taken together, these developments show that periodic Hankel operators form a spectral class parallel to periodic Schrödinger operators but not reducible to them. The Floquet–Bloch picture, spectral bands, and IDS are familiar periodic-operator features; the coexistence of flat and dispersive bands, the possibility of infinite-multiplicity eigenvalues at nonzero energies, and the natural role of logarithmic variables and Mellin analysis are specifically Hankel phenomena (Pastur et al., 29 Sep 2025, Pushnitski et al., 2023).