Abstract
We obtain a complete characterization of the norm attainment set of a bounded linear operator between normed spaces, in terms of semi-inner-product(s) defined on the space. In particular, this answers an open question raised recently in [D. Sain, On the norm attainment set of a bounded linear operator, J. Math. Anal. Appl., 457 (2018), 67–76]. Our results illustrate the applicability of semi-inner-products towards a better understanding of the geometry of normed spaces.
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References
G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J., 1 (1935), 169–172.
J. Chmieliński, Remarks on orthogonality preserving mappings in normed spaces and some stability problems, Banach J. Math. Anal., 1(1) (2007), 117–124.
J. R. Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc., 129 (1967), 436–446.
R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 61 (1947), 265–292.
D. Koehler and P. Rosenthal, On isometries of normed linear spaces, Studia Math., 36 (1970), 213–216.
G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc., 100 (1961), 29–43.
D. Sain, Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, J. Math. Anal. Appl., 447 (2017), 860–866.
D. Sain, On the norm attainment set of a bounded linear operator, J. Math. Anal. Appl., 457 (2018), 67–76.
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Dedicated to Dr. Dwijendra Nath Sain
The research of the author is sponsored by Dr. D. S. Kothari Postdoctoral Fellowship under the mentorship of Professor Gadadhar Misra, to whom the author is immensely indebted. The author feels elated to acknowledge the colossal positive contribution of his alma mater, Ramakrishna Mission Vidyapith, Purulia, in every sphere of his life! He is extremely grateful to Professor Vladimir Kadets for his helpful suggestions towards proving Lemma 2.1 and Professor Kallol Paul for a detailed reading and valuable comments. Last but not the least, the author would like to thank an anonymous referee for his/her fruitful suggestions.
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Sain, D. On the Norm Attainment Set of a Bounded Linear Operator and Semi-Inner-Products in Normed Spaces. Indian J Pure Appl Math 51, 179–186 (2020). https://doi.org/10.1007/s13226-020-0393-9
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DOI: https://doi.org/10.1007/s13226-020-0393-9