Hankel Singular Value Analysis

• Hankel Singular Value Analysis is the study of the singular value spectrum of compact Hankel operators, using Blaschke product invariants to fully characterize their structure.
• This framework enables explicit inverse spectral reconstruction by mapping operator symbols to smooth, finite-dimensional isospectral manifolds.
• It underpins the dynamical analysis of integrable systems, such as the cubic Szegő equation, through action-angle variables and symplectic geometry.

Hankel singular value analysis refers to the study of the singular value spectrum of compact Hankel operators and matrices, with applications spanning operator theory, spectral analysis, integrable systems, and time series/model reduction. This analysis examines the structure of singular values (including their multiplicities and asymptotics), their connection to underlying symbols and functional data, and enables inverse spectral reconstructions as well as the dynamical interpretation of nonlinear flows such as the cubic Szegő equation. The foundational work of Gérard and Grellier provides a complete description of the inverse spectral problem for compact Hankel operators, characterizing them via singular values and Blaschke product invariants, and reveals deep links to symplectic geometry and integrability on associated invariant tori.

1. Spectral Data and Complete Invariants for Compact Hankel Operators

Singular values of a compact Hankel operator $H(u)$ (with symbol $u \in \mathrm{VMO}_+$ on the circle) are the square roots of the eigenvalues of $H^2$, ordered as $s_1 > s_2 > \cdots$ with associated multiplicities. A central result is that $H(u)$ is completely characterized up to unitary equivalence by the collection of its nonzero singular values and a set of finite Blaschke products which serve as angular invariants.

For each block of equal singular value (arising from an eigenvalue of potentially higher multiplicity), the corresponding eigenspace is associated to a monic finite Blaschke product (an inner function), whose degree matches the difference in multiplicities of the eigenvalue in $H^2$ and its shifted companion $K^2$. This structure yields a spectral invariant:

$$\mathcal{S}(u) = \big(\{s_j\}_{j=1}^n,\ \{Y_j\}_{j=1}^n\big)$$

where $Y_j$ are monic Blaschke products (possibly including an $S^1$ phase), and the isospectral set of symbols with given multiplicities $V(d_1,\dots,d_n)$ is parametrized smoothly by these data.

The explicit action of $H(u)$ restricted to such an eigenspace is "twisted" by the Blaschke product: for a singular value $s$ in $EH(u)$ (eigenvalues of $H^2$ with multiplicity one higher than in $K^2$), $H(u)$ acts as multiplication by $s$, modulated by the inner function $Y$:

$H(u)|_{\text{eigenspace}} = s \cdot Y$

This establishes a nonlinear Fourier transform on the space of symbols of compact Hankel operators.

2. Inverse Spectral Theory: Explicit Symbol Reconstruction and Manifold Structure

Given a specified sequence of singular values (with prescribed multiplicities) and associated Blaschke products, the symbol $u$ can be recovered via an explicit determinant formula. In the finite-rank case (with $2q$ nonzero singular values or $2q-1$ if the smallest is zero), $u(z)$ is a finite linear combination of Blaschke products multiplied by minors formed from a matrix determined by the spectral data (see eq. (1.19) in the source).

The mapping $u \mapsto (\{s_j\}, \{Y_j\})$ is a homeomorphism from $V(d_1,\dots,d_n)$ onto an open subset of $\mathbb{R}^n \times$ a product of Blaschke product spaces. Each isospectral set thus forms a finite-dimensional smooth manifold diffeomorphic to $\mathbb{R}^M$. This allows for a geometric interpretation: the space of symbols giving rise to compact Hankel operators with fixed singular data is a smooth foliation into isospectral tori.

3. Dynamical Implications: Cubic Szegő Equation and Action-Angle Variables

The inverse spectral structure is leveraged to analyze the cubic Szegő equation,

$i \partial_t u = \Pi(|u|^2 u)$

where $\Pi$ is the Szegő projector. Under the Szegő flow, the singular values of $H(u)$ and those of its shift-companion $K(u)$ are preserved. The flow of the cubic Szegő equation is explicitly integrable and linearized in the spectral (action-angle) coordinates: each phase parameter in the spectral data evolves as

$Y_r(u(t)) = e^{-i s_r^2 t} Y_r(u(0))$

for each $r$. Solutions with finite $H^{1/2}(S^1)$-norm (finite "momentum") are almost periodic in time with values in $H^{1/2}(S^1)$. This provides a precise dynamical description and quasi-periodicity of solutions.

4. Symplectic Geometry and the Szegő Hierarchy

The analysis extends to the Szegő hierarchy, a family of commuting Hamiltonian flows. The Darboux form of the induced symplectic structure on the isospectral manifold is

$$\omega = \sum_{r=1}^n d\left(\frac{1}{2} s_r^2\right) \wedge d\psi_r$$

where $s_r^2/2$ are the actions, and $\psi_r$ are angle variables from the Blaschke factors. This reduction establishes complete integrability for the finite-dimensional invariant tori formed by the isospectral manifolds $V(d_1,\dots,d_n)$. The involutive property of these manifolds, as well as their preservation under the hierarchy flows, allows for the identification of invariant tori in the phase space of the (nonlinear) evolution.

A deeper geometric interpretation arises from the connection to unitary equivalence classes of operator pairs $(H(u), K(u))$. Two symbols $u$, $\tilde{u}$ are equivalent if there exist unitaries intertwining $H$ and $K$, possibly modulo a phase adjustment depending on spectral projections. The spectral data parametrization precisely classifies these unitary equivalence classes.

5. Auxiliary Formulas, Blaschke Product Structure, and Schmidt Pair Analysis

Several key technical results complement the main theory:

• Bateman-type formulas relate the resolvent of $H^2$ and $K^2$ (or the difference of their resolvents) to infinite products over the singular values. These are instrumental in establishing explicit trace and determinant identities relevant for the inverse spectral map.
• The set of monic finite Blaschke products of degree $d$ is shown to be a smooth manifold, diffeomorphic to $\mathbb{R}^{2d}$ (or $S^1 \times \mathbb{R}^{2d}$ with phase), with further structure provided by roots and phase factor parameterizations. Schur–Cohn criteria characterize when polynomials have all roots in the unit disk.
• Revisiting the Adamyan–Arov–Krein results, the paper clarifies the structure of Schmidt pairs for Hankel operators: every nonzero element of an eigenspace is a product of a finite-dimensional polynomial space and a fixed outer function, and the best low-rank approximation (in norm) is achieved by truncating at singular values, with the error matching the discarded singular value.

6. Broader Impact and Extensions

This framework establishes a full nonlinear spectral theory for compact Hankel operators, allowing for arbitrary singular value multiplicities. The approach resolves inverse spectral problems in non-selfadjoint settings and clarifies the fine structure of time evolution in integrable nonlinear PDEs such as the cubic Szegő equation. The identification of smooth isospectral tori, action-angle coordinates, and the role of Blaschke product invariants opens perspectives for further research in nonlinear spectral theory, the structure of integrable systems, and the analysis of infinite-dimensional Hamiltonian systems.

The techniques unify functional analysis, operator theory, and integrable dynamics, and suggest that similar analytic and geometric strategies may be applicable to related non-normal operators and PDEs admitting nonlinear spectral transforms. The explicit geometric-parametric description via singular value data and inner functions (Blaschke products) is especially powerful for the classification and dynamical analysis of non-selfadjoint operators.