Opuscula Math. 37, no. 1 (2017), 189-218
http://dx.doi.org/10.7494/OpMath.2017.37.1.189

Opuscula Mathematica

# Hankel and Toeplitz operators: continuous and discrete representations

Dmitri R. Yafaev

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

Keywords: unbounded Hankel and Toeplitz operators, various representations, moment problems, generalized Hilbert matrices.

Mathematics Subject Classification: 47B25, 47B35.

Full text (pdf)