Adjoint-Product Commutativity of Little Hankel Operators with Trigonometric Polynomial Symbols on Hardy–Sobolev Spaces

Abstract

This paper studies the algebraic properties of little Hankel operators on Hardy–Sobolev spaces $H_s^2$, focusing on a notion of commutativity defined via adjoint products. For symbols $\phi$ and $\psi$, whose co-analytic parts are trigonometric polynomials, we consider the condition ${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}\mathrm{on}{H}_s^2.$ It is shown that this adjoint-product commutativity holds if and only if the co-analytic parts of the symbols are real scalar multiples of one another. As a consequence, the commutant of a nonzero Hankel operator on $H_s^2$, within the class of Hankel operators whose co-analytic symbols are trigonometric polynomials, is one-dimensional over $\mathbb{R}$. The proof relies on a direct coefficient analysis exploiting the finite Hankel structure induced by polynomial symbols. The result applies uniformly to all Sobolev exponents $s\ge 0$, including the classical Hardy space $s=0$ and the Dirichlet case $s=1/2$.

1. Introduction

Hankel operators are fundamental objects in the study of complex and functional analysis. This is largely due to their connections with Hardy spaces and their role in the structure of certain function spaces. Acting as the co-analytic counterparts to Toeplitz operators, they naturally arise when studying projection onto the unit circle $\mathbb{T}$ and the behavior of multiplication operators within $L^2\left(\mathbb{T}\right)$. For a general Hilbert-space treatment of Hankel operators, see [1]. These operators have been extensively studied in relation to boundedness (see [2]), compactness (see [3]), and approximation properties (see [4]). Related questions of boundedness and compactness for Hankel operators on Bergman spaces, and for Toeplitz and Hankel operators on the Paley–Wiener space, are studied in [5,6]. They continue to attract interest due to their applications in both theoretical mathematics and applied fields such as signal processing and control theory (see [7,8]). Their interplay with Sobolev spaces and other weighted Hardy-type spaces also provides a rich setting for exploring deeper operator-theoretic phenomena.

Hardy–Sobolev spaces provide a natural intermediate framework between classical Hardy spaces and smoother analytic function spaces. They retain the analytic structure of the Hardy space $H^2$ while incorporating derivative-type regularity through Sobolev weights on Fourier coefficients. As a result, these spaces offer a flexible setting in which one may study how operator-theoretic phenomena depend on smoothness parameters. In particular, they contain several classical spaces as special cases, including the Hardy space when $s=0$ and the Dirichlet-type case when $s=1/2$. For general background on Sobolev spaces, function spaces, and their analytic variants, we refer to [9,10].

Commutativity problems for Toeplitz and Hankel operators have a long history in operator theory, dating back to classical foundational work such as that of Brown and Halmos [11]. Later investigations have extensively examined commutativity of Toeplitz operators on various analytic function spaces (see, for example, [12]). Ding and Zheng [13] completely characterized when the commutator of two Toeplitz operators or two Hankel operators on the Hardy space $H^2$ has finite rank. Questions involving commutativity of mixed Toeplitz and Hankel operators have also received considerable attention (see, for example, [14,15]). In addition, essential-commutant characterizations of Hankel operators relative to Toeplitz operators have been obtained by function-algebra methods and interpolating Blaschke products (see [16,17]). More recently, commutativity and related structural properties of Hankel operators themselves have been studied, including dual Hankel operators with harmonic symbols acting on orthogonal complements of analytic function spaces (see [18]). Furthermore, in [19] the authors characterized when three Hankel operators on $H^2$ commute pairwise in terms of linear relations among their symbols. Extensions to multivariable settings have also been considered. For instance, Curto, Datt, and Gupta [20] analyzed commutativity of Toeplitz and Hankel operators on the Hardy space over the n-torus. These contributions typically rely on symbolic decompositions, harmonic extensions, or function-algebra methods. In contrast, the present paper studies Hankel operator commutativity through an adjoint-product commutativity condition (1) formulated directly at the operator level.

A classical little Hankel operator with symbol $\phi$ maps the Hardy space $H^2$ into its orthogonal complement ${\left(H^2\right)}^{\perp }$. Therefore, Hankel operators do not naturally form a closed algebra under composition. This structural limitation complicates the formulation of commutativity conditions, distinguishing Hankel operators from their Toeplitz counterparts. In this paper, we study adjoint-product commutativity for little Hankel operators on Hardy–Sobolev spaces under the assumption that the co-analytic parts of the symbols are trigonometric polynomials of finite degree. This restriction already captures the essential algebraic structure of the problem while allowing a direct coefficient-level analysis. Specifically, we investigate the adjoint-product commutativity condition

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}\mathrm{on}{H}_s^2$$

(1)

where $\phi ,\psi \in L^{\infty }\left(\mathbb{T}\right)$ and $H_{\phi }^{\left(s\right)}$ denotes the little Hankel operator on $H_s^2$ with symbol $\phi$. Our main result shows that a necessary and sufficient condition for (1) to hold on $H_s^2$ is that the co-analytic parts of the symbols $\phi$ and $\psi$ are real scalar multiples of each other, that is,

$${\phi }_{-}=\lambda {\psi }_{-}$$

for some $\lambda \in \mathbb{R}$, where ${\phi }_{-}$ and ${\psi }_{-}$ denote the co-analytic parts of $\phi$ and $\psi$, respectively. This result establishes a rigidity phenomenon: within the class of polynomial symbols, the commutant of a nonzero Hankel operator, with respect to the adjoint-product commutativity condition, is one-dimensional. In particular, within the class of finite-rank Hankel operators, adjoint-product commutativity is rigid in the sense that any Hankel operator that commutes with a given nonzero Hankel operator must be a real scalar multiple of it. This characterization is purely operator-theoretic and does not rely on symbolic decompositions or inner function representations. The result holds uniformly for all Sobolev exponents $s\ge 0$. In particular, it includes the classical Hardy space case $s=0$ and the Dirichlet space case $s=1/2$.

This paper is organized as follows. Section 2 provides background on Hardy–Sobolev spaces and little Hankel operators. It also introduces Hankel operators with $L^{\infty }\left(\mathbb{T}\right)$ symbols acting on $H_s^2$ and formulates the adjoint-product commutativity condition that motivates the main result. In Section 3, we recall basic properties of little Hankel operators on Hardy–Sobolev spaces and fix the notation used throughout the paper. Section 4 presents the main theorem (Theorem 1) and its proof, including a reduction to the co-analytic components and the coefficient-level analysis that leads to the real scalar dependence. Corollary 2 characterizes adjoint-product commutativity for Hankel operators with trigonometric polynomial symbols in terms of the linear dependence of their co-analytic parts. Section 5 provides examples illustrating the main theorem and the necessity of the real scalar condition.

2. Preliminaries

We begin by introducing the notation and the function spaces that will be used throughout this paper. Let $\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}$ denote the unit circle. The space $L^2\left(\mathbb{T}\right)$ is the space of all square-integrable functions on $\mathbb{T}$ with respect to the normalized Lebesgue measure. Let $H^2$ be the classical Hardy space on $\mathbb{T}$. Every function $f\in L^2\left(\mathbb{T}\right)$ has a Fourier series

$$f\left({\mathrm{e}}^{int}\right)=\sum _{n\in \mathbb{Z}}{c}_n{\mathrm{e}}^{int}$$

and the orthogonal projection $P:L^2\left(\mathbb{T}\right)\to H^2$ is given by

$$Pf=\sum _{n\ge 0}{c}_n{\mathrm{e}}^{int}.$$

Thus, every function $f\in L^2\left(\mathbb{T}\right)$ admits the decomposition

$$f=f_{+}+f_{-},$$

where $f_{+}=Pf\in H^2$ and $f_{-}$ contains the negative Fourier coefficients of f.

For $s\ge 0$, the Hardy–Sobolev space $H_s^2$ consists of all functions in $H^2$ whose Fourier series

$$f\left({\mathrm{e}}^{it}\right)=\sum _{n\ge 0}{c}_n{\mathrm{e}}^{int}$$

satisfies

$$\sum _{n\ge 0}{(1+n)}^{2s}{\left|c_n\right|}^2<\infty .$$

Clearly, $H_0^2=H^2$, and $H_s^2$ is a Hilbert space with respect to the Sobolev inner product

$${\langle f,g\rangle }_{H_s^2}=\sum _{n\ge 0}{(1+n)}^{2s}{c}_n\overline{d_n},$$

where $g\left({\mathrm{e}}^{it}\right)={\sum }_{n\ge 0}{d}_n{\mathrm{e}}^{int}$. To place $H_s^2$ in a Hilbert space framework, we consider the weighted space

$$L_s^2\left(\mathbb{T}\right)=\left\{f\left({\mathrm{e}}^{it}\right)=\sum _{n\in \mathbb{Z}}{c}_n{\mathrm{e}}^{int}:\sum _{n\in \mathbb{Z}}{(1+|n\left|\right)}^{2s}{\left|c_n\right|}^2<\infty \right\},$$

equipped with the inner product

$${\langle f,g\rangle }_{L_s^2}=\sum _{n\in \mathbb{Z}}{(1+|n\left|\right)}^{2s}{c}_n\overline{d_n}.$$

Then, $H_s^2$ is a closed subspace of $L_s^2\left(\mathbb{T}\right)$. Let $P_s$ denote the orthogonal projection from $L_s^2\left(\mathbb{T}\right)$ onto $H_s^2$ with respect to the inner product of $L_s^2$. Explicitly, if $f\left({\mathrm{e}}^{it}\right)={\sum }_{n\in \mathbb{Z}}{c}_n{\mathrm{e}}^{int}$, then

$$P_{s}f\left({\mathrm{e}}^{it}\right)=\sum _{n\ge 0}{c}_n{\mathrm{e}}^{int}.$$

Although the formula for $P_s$ coincides with the classical Hardy projection P, orthogonality is taken with respect to the inner product of $L_s^2$, and, hence, $P_s$ depends on the parameter s. The weighted projection $P_s$ preserves the analytic Fourier modes while respecting the Sobolev inner product. It therefore plays the same structural role on $H_s^2$ as the classical Hardy projection P does on $H^2$, and naturally appears in the definition of little Hankel operators on Hardy–Sobolev spaces.

We write $L^{\infty }\left(\mathbb{T}\right)$ for the space of all essentially bounded measurable functions on $\mathbb{T}$, equipped with the norm

$${\parallel \phi \parallel }_{\infty }=\underset{\zeta \in \mathbb{T}}{\mathrm{ess\; sup}}\left|\phi \left(\zeta \right)\right|.$$

For $\phi \in L^{\infty }\left(\mathbb{T}\right)$ with Fourier series

$$\phi \left({\mathrm{e}}^{it}\right)=\sum _{n\in \mathbb{Z}}{a}_n{\mathrm{e}}^{int},$$

we write $\phi ={\phi }_{+}+{\phi }_{-}$ where

$${\phi }_{+}\left({\mathrm{e}}^{it}\right)=\sum _{n\ge 0}{a}_n{\mathrm{e}}^{int}\mathrm{and}{\phi }_{-}\left({\mathrm{e}}^{it}\right)=\sum _{n<0}{a}_n{\mathrm{e}}^{int}.$$

We denote by $H^{\infty }$ the space of bounded analytic functions on the unit disk $\mathbb{D}$, which can be identified with the closed subspace of $L^{\infty }\left(\mathbb{T}\right)$ consisting of functions whose negative Fourier coefficients vanish. In particular, ${\phi }_{+}\in H^{\infty }$ and ${\phi }_{-}\in \overline{H^{\infty }}$.

The classical little Hankel operator with symbol $\phi \in L^{\infty }\left(\mathbb{T}\right)$ is defined by

$$H_{\phi }f=(I-P)\left(\phi f\right).$$

It maps $H^2$ into ${\left(H^2\right)}^{\perp }$. Similarly, define the little Hankel operator $H_{\phi }^{\left(s\right)}:H_s^2\to {\left(H_s^2\right)}^{\perp }$ by

$$H_{\phi }^{\left(s\right)}f=(I-P_s)\left(\phi f\right),$$

where ${\left(H_s^2\right)}^{\perp }$ denotes the orthogonal complement of $H_s^2$ in $L_s^2$. The composition $H_{\phi }^{\left(s\right)}{H}_{\psi }^{\left(s\right)}$ is, in general, not defined on $H_s^2$. Accordingly, we study the adjoint-product commutativity condition (1). This equality will serve as our notion of commutativity for Hankel operators on $H_s^2$. All Hankel operators considered in this paper are little Hankel operators.

3. Basic Properties of Little Hankel Operators

In this section, we collect several preliminary results on little Hankel operators acting on the Hardy–Sobolev spaces $H_s^2$. We first show that such operators depend only on the co-analytic part of the symbol, which allows the problem to be reduced to negative Fourier coefficients. We then characterize when $H_{\phi }^{\left(s\right)}$ is the zero operator and obtain matrix representations of $H_{\phi }^{\left(s\right)}$ and of the adjoint product ${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}$ with respect to natural orthonormal bases. These formulas will play a central role in the proof of the main theorem.

Lemma 1.

For every symbol $\phi \in L^{\infty }\left(\mathbb{T}\right)$ and every $s\ge 0$, the Hankel operator on $H_s^2$ satisfies

$$H_{\phi }^{\left(s\right)}=H_{{\phi }_{-}}^{\left(s\right)}.$$

Proof. 

Take $f\in H_s^2\subset H^2$. Then, ${\phi }_{+}f$ has no negative Fourier coefficients, hence

$$(I-P_s)\left({\phi }_{+}f\right)=0.$$

Therefore,

$$H_{\phi }^{\left(s\right)}f=(I-P_s)\left(\phi f\right)=(I-P_s)\left({\phi }_{-}f\right)=H_{{\phi }_{-}}^{\left(s\right)}f,$$

for all $f\in H_s^2$, so $H_{\phi }^{\left(s\right)}=H_{{\phi }_{-}}^{\left(s\right)}$. □

Corollary 1.

The Hankel operator $H_{\phi }^{\left(s\right)}$ equals zero if and only if ${\phi }_{-}=0$.

Proof. 

Assume first that ${\phi }_{-}=0$. Then, by Lemma 1, for every $f\in H_s^2$,

$$H_{\phi }^{\left(s\right)}f=H_{{\phi }_{-}}^{\left(s\right)}f=H_0^{\left(s\right)}f=0.$$

Conversely, suppose ${\phi }_{-}\ne 0$. Then,

$$H_{{\phi }_{-}}^{\left(s\right)}1=(I-P_s){\phi }_{-}1\ne 0,$$

since ${\phi }_{-}\notin H_s^2$. Thus, $H_{{\phi }_{-}}^{\left(s\right)}\ne 0$, and, therefore, $H_{\phi }^{\left(s\right)}\ne 0$. □

Proposition 1.

Let $s\ge 0$ and $\phi ,\psi \in L^{\infty }\left(\mathbb{T}\right)$. Write

$${\phi }_{-}\left({\mathrm{e}}^{it}\right)=\sum _{k\ge 1}{a}_k{\mathrm{e}}^{-ikt},{\psi }_{-}\left({\mathrm{e}}^{it}\right)=\sum _{k\ge 1}{b}_k{\mathrm{e}}^{-ikt}.$$

Set

$$e_n^{\left(s\right)}\left({\mathrm{e}}^{it}\right)={(1+n)}^{-s}{\mathrm{e}}^{int},n\ge 0,$$

which forms an orthonormal basis of $H_s^2$, and

$$f_m^{\left(s\right)}\left({\mathrm{e}}^{it}\right)={(1+m)}^{-s}{\mathrm{e}}^{-i(m+1)t},m\ge 0,$$

which forms an orthonormal basis of ${\left(H_s^2\right)}^{\perp }$ (with respect to the Sobolev inner product). Then, for every $n\ge 0$,

$$H_{\phi }^{\left(s\right)}{e}_n^{\left(s\right)}=\sum _{m\ge 0}{a}_{m+n+1}{(1+m)}^s{(1+n)}^{-s}{f}_m^{\left(s\right)}.$$

Consequently, the matrix of the adjoint-product ${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}$ with respect to the basis ${\left\{e_n^{\left(s\right)}\right\}}_{n\ge 0}$ is given by

$${〈{\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}{e}_n^{\left(s\right)},e_k^{\left(s\right)}〉}_{H_s^2}={(1+n)}^{-s}{(1+k)}^{-s}\sum _{m\ge 0}{(1+m)}^{2s}\overline{a_{m+k+1}}{b}_{m+n+1}.$$

Proof. 

By Lemma 1, we may replace $\phi$ and $\psi$ by their negative parts. Fix $n\ge 0$. Using the Fourier expansions above, we have

$${\phi }_{-}\left({\mathrm{e}}^{it}\right)e_n^{\left(s\right)}\left({\mathrm{e}}^{it}\right)={(1+n)}^{-s}\sum _{\ell \ge 1}{a}_{\ell }{\mathrm{e}}^{i(n-\ell )t}.$$

The negative Fourier terms correspond to $\ell \ge n+1$. Writing $\ell =m+n+1$ with $m\ge 0$, we obtain

$$(I-P_s)\left({\phi }_{-}{e}_n^{\left(s\right)}\right)={(1+n)}^{-s}\sum _{m\ge 0}{a}_{m+n+1}{\mathrm{e}}^{-i(m+1)t}.$$

Expressing ${\mathrm{e}}^{-i(m+1)t}={(1+m)}^s{f}_m^{\left(s\right)}$, we obtain

$$H_{\phi }^{\left(s\right)}{e}_n^{\left(s\right)}=\sum _{m\ge 0}{a}_{m+n+1}{(1+m)}^s{(1+n)}^{-s}{f}_m^{\left(s\right)}.$$

Similarly,

$$H_{\psi }^{\left(s\right)}{e}_n^{\left(s\right)}=\sum _{m\ge 0}{b}_{m+n+1}{(1+m)}^s{(1+n)}^{-s}{f}_m^{\left(s\right)},$$

and

$$H_{\phi }^{\left(s\right)}{e}_k^{\left(s\right)}=\sum _{m\ge 0}{a}_{m+k+1}{(1+m)}^s{(1+k)}^{-s}{f}_m^{\left(s\right)}.$$

Hence, by the definition of the adjoint,

$${〈{\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}{e}_n^{\left(s\right)},e_k^{\left(s\right)}〉}_{H_s^2}={〈H_{\psi }^{\left(s\right)}{e}_n^{\left(s\right)},H_{\phi }^{\left(s\right)}{e}_k^{\left(s\right)}〉}_{{\left(H_s^2\right)}^{\perp }}.$$

Substituting the above expansions and using the orthonormality of ${\left\{f_m^{\left(s\right)}\right\}}_{m\ge 0}$, we obtain

$${〈{\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}{e}_n^{\left(s\right)},e_k^{\left(s\right)}〉}_{H_s^2}={(1+n)}^{-s}{(1+k)}^{-s}\sum _{m\ge 0}{(1+m)}^{2s}\overline{a_{m+k+1}}{b}_{m+n+1},$$

which proves the stated matrix formula. □

Proposition 2.

Let $s\ge 0$ and $\phi ,\psi \in L^{\infty }\left(\mathbb{T}\right)$. Write

$${\phi }_{-}\left({\mathrm{e}}^{it}\right)=\sum _{j\ge 1}{a}_j{\mathrm{e}}^{-ijt},{\psi }_{-}\left({\mathrm{e}}^{it}\right)=\sum _{j\ge 1}{b}_j{\mathrm{e}}^{-ijt}.$$

Then, the adjoint-product commutativity condition

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}$$

holds on $H_s^2$, if and only if for all $n,k\ge 0$,

$$\sum _{m\ge 0}{(1+m)}^{2s}\overline{a_{m+k+1}}{b}_{m+n+1}=\sum _{m\ge 0}{(1+m)}^{2s}\overline{b_{m+k+1}}{a}_{m+n+1}.$$

Proof. 

By Proposition 1, the $(n,k)$ entry of the matrix of the adjoint product ${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}$ is given by

$${(1+n)}^{-s}{(1+k)}^{-s}\sum _{m\ge 0}{(1+m)}^{2s}\overline{a_{m+k+1}}{b}_{m+n+1}.$$

Similarly, the $(n,k)$ entry of the matrix of ${\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}$ is

$${(1+n)}^{-s}{(1+k)}^{-s}\sum _{m\ge 0}{(1+m)}^{2s}\overline{b_{m+k+1}}{a}_{m+n+1}.$$

Hence, the equality of the operators is equivalent to the equality of all $(n,k)$ entries. □

4. Main Results

In this section, we restrict our attention to symbols whose co-analytic parts are trigonometric polynomials of a finite degree. In this case, the matrix representation in Proposition 2 reduces to a finite system of equations, which allows a coefficient-level analysis of the commutativity condition.

Theorem 1.

Let $s\ge 0$ and let $\phi ,\psi \in L^{\infty }\left(\mathbb{T}\right)$. Assume that the co-analytic parts ${\phi }_{-}$ and ${\psi }_{-}$ are trigonometric polynomials of a finite degree. Then, the Hankel operators satisfy

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}\mathrm{on}{H}_s^2$$

if and only if there exists a real constant $\lambda \in \mathbb{R}$, such that

$${\phi }_{-}=\lambda {\psi }_{-}.$$

Proof. 

If ${\psi }_{-}=0$, then by Corollary 1 we have $H_{\psi }^{\left(s\right)}=0$. Hence,

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}=0.$$

Moreover, the relation ${\phi }_{-}=\lambda {\psi }_{-}$ reduces to ${\phi }_{-}=0$, so the conclusion is immediate in this trivial case. We therefore assume throughout the rest of the proof that ${\psi }_{-}\ne 0$.

Suppose that the commutativity condition holds. Write

$${\phi }_{-}\left({\mathrm{e}}^{it}\right)={\displaystyle \sum _{j=1}^N}{a}_j{\mathrm{e}}^{-ijt},{\psi }_{-}\left({\mathrm{e}}^{it}\right)={\displaystyle \sum _{j=1}^N}{b}_j{\mathrm{e}}^{-ijt}.$$

where $N=\max \{\mathrm{deg}{\phi }_{-},\mathrm{deg}{\psi }_{-}\}$, and the missing coefficients are taken to be zero. By Lemma 1, $H_{\phi }^{\left(s\right)}$ and $H_{\psi }^{\left(s\right)}$ depend only on ${\phi }_{-}$ and ${\psi }_{-}$, respectively. Hence, we may assume that $\phi ={\phi }_{-}$ and $\psi ={\psi }_{-}$. By Proposition 2, the hypothesis is equivalent to the identity

$${\displaystyle \sum _{m=0}^{N-\max \{n,k\}-1}}{(1+m)}^{2s}\overline{a_{m+k+1}}{b}_{m+n+1}={\displaystyle \sum _{m=0}^{N-\max \{n,k\}-1}}{(1+m)}^{2s}\overline{b_{m+k+1}}{a}_{m+n+1}$$

(2)

for all $0\le n,k\le N-1$. Let r be the largest index, such that $b_r\ne 0$. Such an index exists since ${\psi }_{-}\ne 0$. Then, $b_j=0$ for all $j>r$. We first show that $a_j=0$ for all $j>r$. Suppose not, and let ℓ be the largest index, such that $a_{\ell }\ne 0$. Taking $k=\ell -1$ and $n=r-1$ in (2), only the term $m=0$ survives, and we obtain

$$\overline{a_{\ell }}{b}_r=0.$$

Since $b_r\ne 0$, this implies that $a_{\ell }=0$, a contradiction. Therefore, $a_j=0$ for all $j>r$.

Now, taking $k=r-1$ in (2), we obtain

$$\overline{a_r}{b}_{n+1}=\overline{b_r}{a}_{n+1},0\le n\le r-1.$$

(3)

In particular, for $n=r-1$, we obtain

$$\overline{a_r}{b}_r=\overline{b_r}{a}_r,$$

so $\overline{a_r}{b}_r\in \mathbb{R}$. Since $b_r\ne 0$, we may define $\lambda :={\displaystyle \frac{a_r}{b_r}}\in \mathbb{R}$. Substituting $a_r=\lambda b_r$ into Equation (3) yields

$$\lambda \overline{b_r}{b}_{n+1}=\overline{b_r}{a}_{n+1},0\le n\le r-1,$$

and, hence,

$$a_{n+1}=\lambda b_{n+1}0\le n\le r-1.$$

Therefore,

$$a_j=\lambda b_{j}1\le j\le r.$$

Hence, $a_j=\lambda b_j$ for all $1\le j\le N$, and so

$${\phi }_{-}=\lambda {\psi }_{-}.$$

Conversely, If ${\phi }_{-}=\lambda {\psi }_{-}$ for some $\lambda \in \mathbb{R},$ then by Lemma 1,

$$H_{\phi }^{\left(s\right)}=H_{{\phi }_{-}}^{\left(s\right)}=\lambda H_{{\psi }_{-}}^{\left(s\right)}=\lambda H_{\psi }^{\left(s\right)}.$$

Therefore,

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(\lambda H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}=\lambda {\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)},$$

and similarly,

$${\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}\lambda H_{\psi }^{\left(s\right)}=\lambda {\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}.$$

Hence,

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}.$$

This completes the proof. □

The conclusion ${\phi }_{-}=\lambda {\psi }_{-}$ with $\lambda \in \mathbb{R}$ reflects the fact that our commutativity condition is formulated in terms of adjoint products. Indeed, if $H_{\phi }^{\left(s\right)}=\lambda H_{\psi }^{\left(s\right)}\ne 0$, then

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}=\overline{\lambda }{\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)},{\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}=\lambda {\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}.$$

Hence, the equality of these two operators forces $\overline{\lambda }=\lambda$, that is, $\lambda \in \mathbb{R}$. Thus, the adjoint-product commutativity condition admits only a real proportionality constant.

Corollary 2.

Let $s\ge 0$ and let $\phi ,\psi \in L^{\infty }\left(\mathbb{T}\right)$. Assume that the co-analytic parts ${\phi }_{-}$ and ${\psi }_{-}$ are trigonometric polynomials of a finite degree, where ${\psi }_{-}\ne 0$. If $H_{\phi }^{\left(s\right)}$ is a Hankel operator, such that

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}\mathrm{on}{H}_s^2,$$

then there exists a real constant λ, such that

$$H_{\phi }^{\left(s\right)}=\lambda H_{\psi }^{\left(s\right)}.$$

Consequently, the commutant of $H_{\psi }^{\left(s\right)}$ within the Hankel class associated with such symbols is one-dimensional.

Proof. 

By Theorem 1, the adjoint-product commutativity condition implies

$${\phi }_{-}=\lambda {\psi }_{-}\mathrm{for}\mathrm{some}\lambda \in \mathbb{R}.$$

Hence

$$H_{\phi }^{\left(s\right)}=H_{{\phi }_{-}}^{\left(s\right)}=H_{\lambda {\psi }_{-}}^{\left(s\right)}=\lambda H_{{\psi }_{-}}^{\left(s\right)}=\lambda H_{\psi }^{\left(s\right)}.$$

 □

As an immediate consequence of Corollary 2, if $H_{\phi }^{\left(s\right)}$ is a finite-rank Hankel operator (equivalently, if ${\phi }_{-}$ is a trigonometric polynomial of a finite degree), then its commutant within the class of Hankel operators with trigonometric polynomial co-analytic symbols is one-dimensional. Since Hankel operators with trigonometric polynomial symbols are finite-rank and, hence, compact, Corollary 2 characterizes the commutant of finite-rank Hankel operators in this class.

5. Examples

In this section, we present several examples illustrating the main result. The first example shows that the adjoint-product commutativity condition (1) holds when the co-analytic parts of the symbols are real scalar multiples of each other. The second demonstrates that a non-real scalar multiple does not satisfy the condition. The final example considers symbols whose co-analytic parts are not scalar multiples, showing that the condition fails in that case.

Example 1.

Let

$${\phi }_{-}\left(z\right)=r(z^{-1}+z^{-2}),{\psi }_{-}\left(z\right)=z^{-1}+z^{-2},$$

where $r\in \mathbb{R}$. Then, ${\phi }_{-}=r{\psi }_{-}$. If $r=0$, then

$${\phi }_{-}=0,$$

and, hence, the adjoint-product commutativity condition (1) trivially holds. If $r\ne 0$, then by Theorem 1,

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}.$$

Hence, the condition (1) holds for any $r\in \mathbb{R}$.

This example illustrates that real proportionality is sufficient for adjoint-product commutativity, whereas the following example shows that complex proportionality alone is not sufficient.

Example 2.

Let

$${\phi }_{-}\left(z\right)=i(z^{-1}+2z^{-2}),{\psi }_{-}\left(z\right)=z^{-1}+2z^{-2}.$$

Then, ${\phi }_{-}=\lambda {\psi }_{-}$, where $\lambda =i\notin \mathbb{R}$. We show that the adjoint-product commutativity does not hold in this case. Indeed,

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{{\phi }_{-}}^{\left(s\right)}\right)}^{*}{H}_{{\psi }_{-}}^{\left(s\right)}={\left(H_{i{\psi }_{-}}^{\left(s\right)}\right)}^{*}{H}_{{\psi }_{-}}^{\left(s\right)}=-i{\left(H_{{\psi }_{-}}^{\left(s\right)}\right)}^{*}{H}_{{\psi }_{-}}^{\left(s\right)}.$$

On the other hand,

$${\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}={\left(H_{{\psi }_{-}}^{\left(s\right)}\right)}^{*}{H}_{{\phi }_{-}}^{\left(s\right)}={\left(H_{{\psi }_{-}}^{\left(s\right)}\right)}^{*}{H}_{i{\psi }_{-}}^{\left(s\right)}=i{\left(H_{{\psi }_{-}}^{\left(s\right)}\right)}^{*}{H}_{{\psi }_{-}}^{\left(s\right)},$$

showing that the adjoint-product commutativity condition fails when the proportionality constant λ is not real.

Example 3.

Let

$${\phi }_{-}\left(z\right)=z^{-1}+2z^{-2}+z^{-3},{\psi }_{-}\left(z\right)=z^{-1}+2z^{-2}-z^{-3}.$$

Then there is no real number r, such that

$${\phi }_{-}=r{\psi }_{-}.$$

Therefore, by Theorem 1, the adjoint-product commutativity condition does not hold. Hence

$${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}\ne {\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}.$$

Remark 1.

In the classical case $s=0$, the Hardy–Sobolev space $H_0^2$ coincides with the Hardy space $H^2$. The orthonormal bases become

$$e_n^{\left(0\right)}\left({\mathrm{e}}^{it}\right)={\mathrm{e}}^{int},f_m^{\left(0\right)}\left({\mathrm{e}}^{it}\right)={\mathrm{e}}^{-i(m+1)t},$$

for $n,m\ge 0$. Moreover, the weighted factors in Proposition 2 disappear, and the adjoint-product commutativity condition reduces to

$$\sum _{m\ge 0}\overline{a_{m+k+1}}{b}_{m+n+1}=\sum _{m\ge 0}\overline{b_{m+k+1}}{a}_{m+n+1},n,k\ge 0.$$

Hence, Theorem 1 yields the corresponding characterization on the classical Hardy space $H^2$.

Similarly, in the case $s=1/2$, corresponding to the Dirichlet-type case, the orthonormal bases become

$$e_n^{(1/2)}\left({\mathrm{e}}^{it}\right)={(1+n)}^{-1/2}{\mathrm{e}}^{int},f_m^{(1/2)}\left({\mathrm{e}}^{it}\right)={(1+m)}^{-1/2}{\mathrm{e}}^{-i(m+1)t},$$

and Theorem 1 remains valid without change.

6. Concluding Remarks

We have shown that, for Hankel operators whose co-analytic symbols are trigonometric polynomials of finite degree (and, hence, finite rank), the adjoint-product commutativity condition (1) forces a real-linear dependence between the co-analytic parts of the symbols. This result is independent of the Sobolev parameter s. This rigidity phenomenon suggests several directions for future investigations, including extensions to more general symbol classes, weighted settings, and higher-dimensional spaces. An interesting open problem is whether a similar rigidity result holds for general $L^{\infty }\left(\mathbb{T}\right)$ symbols without the finite-degree assumption.