Hankel and Toeplitz operators: continuous and discrete representations

TL;DR

This paper establishes a unitary equivalence between continuous and discrete Hankel operators, enabling comprehensive spectral analysis and linking Toeplitz operator representations across different function spaces.

Contribution

It introduces a relation that unifies the spectral analysis of Hankel operators in sequence and function spaces, extending classical results and connecting Toeplitz representations.

Findings

Spectral results for unbounded Hankel operators in ℓ²(Z₊)

Unitary equivalence between sequence and function space Hankel operators

Link between Toeplitz operator representations in different spaces

Abstract

We find a relation guaranteeing that Hankel operators realized in the space of sequences $\ell^2 ({\Bbb Z}_{+}) $ and in the space of functions $L^2 ({\Bbb R}_{+}) $ are unitarily equivalent. This allows us to obtain exhaustive spectral results for two classes of unbounded Hankel operators in the space $\ell^2 ({\Bbb Z}_{+}) $ generalizing in different directions the classical Hilbert matrix. We also discuss a link between representations of Toeplitz operators in the spaces $\ell^2 ({\Bbb Z}_{+}) $ and $L^2 ({\Bbb R}_{+}) $.