Journal of Convex Analysis 29 (2022), No. 1, 165--181
Copyright Heldermann Verlag 2022
Some Remarks on Orthogonality of Bounded Linear Operators
Anubhab Ray
Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India
anubhab.jumath@gmail.com
Kallol Paul
Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India
kalloldada@gmail.com
Debmalya Sain
Department of Mathematics, Indian Institute of Science, Bengaluru 560012, Karnataka, India
saindebmalya@gmail.com
Subhrajit Dey
Department of Mathematics, Muralidhar Girls' College, Kolkata 700029, West Bengal, India
subhrajitdeyjumath@gmail.com
[Abstract-pdf]
We explore the relation between the orthogonality of bounded linear operators in the space of operators and that of elements in the ground space. To be precise, we study if $T, A \in \mathbb{L}(\mathbb{X}, \mathbb{Y})$ satisfy $T \bot_B A$, then whether there exists $ x \in \mathbb{X} $ such that $Tx\bot_B Ax$ with $\|x\| =1$, $\|Tx\| = \|T\|$, where $\mathbb{X}, \mathbb{Y} $ are normed linear spaces. In this context, we introduce the notion of Property $P_n$ for a Banach space and illustrate its connection with orthogonality of a bounded linear operator between Banach spaces. We further study Property $P_n$ for various polyhedral Banach spaces.
Keywords: Orthogonality, linear operators, norm attainment, polyhedral Banach spaces.
MSC: 46B20; 47L05.
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