Opuscula Math. 37, no. 1 (2017), 189-218

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.

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Cite this article as:
Dmitri R. Yafaev, Hankel and Toeplitz operators: continuous and discrete representations, Opuscula Math. 37, no. 1 (2017), 189-218, http://dx.doi.org/10.7494/OpMath.2017.37.1.189

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