Opuscula Math. 38, no. 4 (2018), 573-590
https://doi.org/10.7494/OpMath.2018.38.4.573

Opuscula Mathematica

# Toeplitz versus Hankel: semibounded operators

Dmitri R. Yafaev

Abstract. Our goal is to compare various results for Toeplitz $$T$$ and Hankel $$H$$ operators. We consider semibounded operators and find necessary and sufficient conditions for their quadratic forms to be closable. This property allows one to define $$T$$ and $$H$$ as self-adjoint operators under minimal assumptions on their matrix elements. We also describe domains of the closed Toeplitz and Hankel quadratic forms.

Keywords: semibounded Toeplitz, Hankel and Wiener-Hopf operators, closable and closed quadratic forms.

Mathematics Subject Classification: 47B25, 47B35.

Full text (pdf)

1. N. Akhiezer, The classical moment problem and some related questions in analysis, Oliver and Boyd, Edinburgh and London, 1965.
2. M.Sh. Birman, M.Z. Solomyak, Spectral Theory of Selfadjoint Operators in Hilbert Space, D. Reidel, Dordrecht, 1987.
3. A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, Springer-Verlag, 2006.
4. T. Carleman, Sur les équations intégrales singulières, Uppsala, 1923.
5. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. 1, 2, McGraw-Hill, New York-Toronto-London, 1953.
6. I.M. Gel'fand, G.E. Shilov, Generalized Functions, vol. 1, Academic Press, New York and London, 1964.
7. I. Gohberg, S. Goldberg, M. Kaashoek, Classes of Linear Operators, vol. 1, Birkhäuser, 1990.
8. H. Hamburger, Über eine Erweiterung des Stieltjesschen Momentenproblems, Math. Ann. 81 (1920), 235-319; 82 (1921), 120-164 and 168-187.
9. P. Hartman, On unbounded Toeplitz matrices, Amer. J. Math. 85 (1963) 1, 59-78.
10. K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Inc., Englewood Cliffs, New York, 1962.
11. O.S. Ivašëv-Musatov, On the Fourier-Stieltjes coefficients of singular functions, Dokl. Akad. Nauk SSSR 82 (1952), 9-11 [in Russian].
12. Z. Nehari, On bounded bilinear forms, Ann. Math. 65 (1957), 153-162.
13. N.K. Nikolski, Operators, functions, and systems: an easy reading, vol. I: Hardy, Hankel, and Toeplitz, Math. Surveys and Monographs vol. 92, Amer. Math. Soc., Providence, Rhode Island, 2002.
14. V.V. Peller, Hankel Operators and Their Applications, Springer Verlag, 2002.
15. M. Reed, B. Simon, Methods of modern mathematical physics, vol. 2, Academic Press, San Diego, CA, 1975.
16. D. Sarason, Unbounded Toeplitz operators, Integral Equations Operator Theory 61 (2008), 281-298.
17. H. Widom, Hankel matrices, Trans. Amer. Math. Soc. 121 (1966), 1-35.
18. D.R. Yafaev, Quasi-Carleman operators and their spectral properties, Integral Equations Operator Theory 81 (2015), 499-534.
19. D.R. Yafaev, Unbounded Hankel operator and moment problems, Integral Equations Operator Theory 85 (2016), 289-300.
20. D.R. Yafaev, On semibounded Toeplitz operators, J. Operator Theory 77 (2017), 742-762.
21. D.R. Yafaev, On semibounded Wiener-Hopf operators, Journal of LMS 95 (2017), 101-112.
• Dmitri R. Yafaev
• Université de Rennes I, IRMAR, Campus de Beaulieu, 35042 Rennes Cedex, France
• SPGU, Univ. Nab. 7/9, Saint Petersburg, 199034 Russia
• Communicated by P.A. Cojuhari.
• Accepted: 2017-12-10.
• Published online: 2018-04-11.