Multivariate analogue of slant Toeplitz operators

This paper discusses several structural and fundamental properties of the $k^{th}$-order slant Toeplitz operators on the Lebesgue space of the $n$- torus $\mathbb{T}^n$, for integers $k\geq 2$ and $n\geq 1$. We obtain certain equivalent conditions for the commutativity and essential commutativity of these operators. In the last section, we deal with the spectrum of a $k^{th}$-order slant Toeplitz operator on $L^2(\mathbb{T}^n)$ and investigate the conditions for such an operator to be an isometry, hyponormal or normal.

Keywords:

commutativity, Lebesgue space slant Toeplitz operators, n-dimensional torus,

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[1] S.C. Arora and R. Batra, Generalized slant Toeplitz operators on $H^2$, Math. Nachr. 278 (4), 347-355, 2005.
[2] G. Datt and N. Ohri, Properties of slant Toeplitz operators on the torus, Malays. J. Math. Sci. 12, (2), 195-206, 2018.
[3] G. Datt and S.K. Pandey, Slant Toeplitz operators on Lebesgue space of n-dimensional Torus, Hokkaido Math. J. 49 (3), 363-389, 2020.
[4] X. Ding, S. Sun and D. Zheng, Commuting Toeplitz operators on the bidisk, J. Funct. Anal. 263, 3333-3357, 2012.
[5] C. Gu and D. Zheng, The semi-commutator of Toeplitz operators on the bidisc, J. Operator Theory 38, 173-193, 1997.
[6] H. Guediri, Dual Toeplitz operators on the sphere, Acta Math. Sin. (Engl. Ser.) 19 (9), 1791-1808, 2013.
[7] M.C. Ho, Spectral properties of slant Toeplitz operators, Ph.D. thesis, Purdue- University, Indiana, 1996.
[8] M.C. Ho, Spectra of slant Toeplitz operators with continuous symbol, Michigan Math. J. 44, 157-166, 1997.
[9] Y.F. Lu and B. Zhang, Commuting Hankel and Toeplitz operators on the Hardy space of the bidisk, J. Math. Res. Exposition 30 (2), 205-216, 2010.
[10] V. Peller, Hankel operators and applications, Springer-Verlag, New York, 2003.
[11] W. Rudin, Function Theory in Polydisc, W.A. Benjamin Inc., New York-Amsterdam 1969.
[12] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1971.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

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