Generalized Foguel-Hankel Operators

In this paper we introduce a more general class of Foguel-Hankel operators, where the unilateral shift on $\ell^2(\mathbb{N}) $ is replaced by a general multiplication operator on the Hardy space $H^2$ . We prove that Peller’s condition is sufficient for the operator to be power bounded, but in general it is not necessary. When the Hankel matrix is the Hilbert matrix, we prove that being similar to a contraction is equivalent to the (a priori) weaker Kreiss condition.

💡 Research Summary

The paper introduces a broad generalization of the classical Foguel‑Hankel operators by replacing the unilateral shift S on ℓ²(ℕ) (or equivalently the multiplication by the coordinate function on the Hardy space H²) with an arbitrary bounded analytic multiplication operator M_φ, where φ is a holomorphic self‑map of the unit disc. The resulting 2×2 block operator acting on H²⊕H² is denoted Γ_{f,φ} =