Multivariate analogue of slant Toeplitz operators

Year 2021, Volume: 50 Issue: 3, 678 - 691, 07.06.2021

Gopal Datt , Shesh Pandey

https://doi.org/10.15672/hujms.663262

Cited By: 1

Abstract

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

References

• [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.