Abstract
We study the norm attainment set and the minimum norm attainment set of an operator defined on \(\ell _p^2 ({\mathbb {R}})\,(1<p<\infty )\). We obtain a complete description of the same from the point of view of cardinality. In particular, we establish the optimal bounds in both the cases. Whenever the operator is not a scalar multiple of an isometry, the upper bound is 4. Moreover, we obtain an upper bound for the norm of an operator on \(\ell _p^2({\mathbb {R}})\,(1<p<\infty )\). We further prove that there is no non-zero left symmetric operator on \(\ell _p^2({\mathbb {C}})\,(2\le p<\infty )\).
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Acknowledgements
The present improved version of the paper is due to the suggestions and comments from the referee for which authors would like to extend their heartfelt thanks. Mr. Kalidas Mandal would like to thank CSIR, Govt. of India for the financial support in the form of Senior Research Fellowship under the mentorship of Prof. Kallol Paul. Dr. Debmalya Sain feels elated to acknowledge the motivating presence of his childhood friend Dr. Sourav Saha, an eminent biologist, in his life. Miss Arpita Mal would like to thank UGC, Govt. of India for the financial support in the form of Senior Research Fellowship under the mentorship of Prof. Kallol Paul. The research of Prof. Kallol Paul is supported by project MATRICS (Ref. no. MTR/2017/000059), SERB, DST, Govt. of India.
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Communicated by Miguel Martin.
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Mandal, K., Sain, D., Mal, A. et al. Norm attainment set and symmetricity of operators on \(\ell _p^2\). Adv. Oper. Theory 7, 3 (2022). https://doi.org/10.1007/s43036-021-00167-w
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DOI: https://doi.org/10.1007/s43036-021-00167-w