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

This paper studies the algebraic properties of little Hankel operators on Hardy–Sobolev spaces Hs2, focusing on a notion of commutativity defined via adjoint products. For symbols φ and ψ, whose co-analytic parts are trigonometric polynomials, we consider the condition Hφ(s)*Hψ(s)=Hψ(s)*Hφ(s)onHs2. 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 Hs2, within the class of Hankel operators whose co-analytic symbols are trigonometric polynomials, is one-dimensional over 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≥0, including the classical Hardy space s=0 and the Dirichlet case s=1/2.