Boundedness of a new kind of Toeplitz operator on $2\pi$-periodic holomorphic functions on the upper halfplane

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, University of Kurdistan, Sanandaj, Iran

2 Department of mathematics, Faculty of Science, University of Kurdistan, Sanandaj, Iran

Abstract

In the present paper, We introduce a new kind of Toeplitz operators on the spaces of $2\pi$ periodic holomorphic functions on the upper halfplane equipped with an integral norm similar to the norm of $L^{p}$ spaces. We prove the boundedness of Toeplitz operators in the case of bounded symbols. Also, we state some open problems for unbounded symbols and other cases in which our spaces are not Hilbert spaces.

Keywords

[1] M.A. Ardalani, On the isomorphism classification of spaces 2π periodic holomorphic functions on the upper halfplane, J. Math. Anal. Appl. 459 (2018) 350–359.
[2] O. Constantin and J. A. Pelaez, Boudndedness of the Bergman projection on Lp-spaces with expotential weights, Bull. Sci. Math. 139 (2015) 245–268.
[3] M. Dostanic, Unboundedness of the Bergman projection on Lp spaces with expotential weights, Proc. Edinb. Math. Soc. 47 (2004) 111–117.
[4] M. Englis, Toeplitz operators and weighted Bergman kernels, J. Funct. Anal. 255 (6) (2008) 1419–1457.
[5] S. Grudsky, A. Karapetyants, N. Vasilevski, Toeplitz operators on the unit ball of CN with radial symbols, J. Oper. Theory 49 (2003) 325–346.
[6] W. Lusky and J. Taskinen, Toeplitz operators on Bergman spaces and Hardy multipliers, Studia. Math. 204 (2011) 137–154.
[7] K. Zhu, Operator Theory in Function Spaces. 2nd edition, Mathematical surves and monographs, Vol. 138, American Mathematical Society, Providence, RI, 2007.
Volume 13, Issue 1
March 2022
Pages 2665-2670
  • Receive Date: 05 January 2021
  • Accept Date: 19 February 2021