Author(s)

Zakieldeen Aboabuda Mohamed Alhassn Ali¹, Amna Mahmoud Ahmed Bakhit¹, Isra Abdalhleem Hassan Ali¹, Manal Yagoub Ahmed Juma², Tahani Mahmoud Ahmed Mohamed¹, Sumia Izzeldeen Abdallah Abdelmajeed³

¹Deanship of the Preparatory Year, Prince Sattam Bin Abdulaziz University, Alkharj, Saudi Arabia.
²Department of Mathematic, Faculty of Science, University of Qassim, Buraidah, Saudi Arabia.
³Department of Mathematic, Gezira Collage of Technology, Khartoum, Sudan.

Important operator characteristics (such boundedness or compactness) for particular classes of operators on particular reproducing kernel Hilbert spaces may be impacted by the behaviour of the operators on the reproducing kernels. These results have been shown for Toeplitz operators on the Paley-Wiener space, a reproducing kernel Hilbert space over C. Furthermore, we show how the norm of such an operator has no relation to the supremum of the norms of the pictures of the normalization reproducing kernels of the space. As a result, if this supremum is finite, the operator is implicitly bounded. To further demonstrate that the operator norm is not the same as the supremum of the norms of the pictures of the real normalized reproducing kernels, another example is also provided. We also set out a necessary and sufficient condition for the operators’ compactness in terms of their limiting function on the reproducing kernels.

Toeplitz Operators, Compact Operator, The Reproducing Kernel

Ali, Z. , Bakhit, A. , Ali, I. , Juma, M. , Mohamed, T. and Abdelmajeed, S. (2023) Kernel Thesis Reproduction and Truncated Toeplitz Operators. Journal of Applied Mathematics and Physics, 11, 4016-4026. doi: 10.4236/jamp.2023.1112257.