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arXiv · 2104.08488

On approximate orthogonality and symmetry of operators in semi-Hilbertian structure

Jeet Sen·Debmalya Sain·Kallol Paul
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Abstract

The purpose of the article is to generalize the concept of approximate Birkhoff-James orthogonality, in the semi-Hilbertian structure. Given a positive operator $ A $ on a Hilbert space $ \mathbb{H}, $ we define $ (\epsilon,A)- $approximate orthogonality and $ (\epsilon,A)- $approximate orthogonality in the sense of Chmieli$\acute{n}$ski and establish a relation between them. We also characterize $ (\epsilon,A)- $approximate orthogonality in the sense of Chmieli$\acute{n}$ski for $A$-bounded and $A$-bounded compact operators. We further generalize the concept of right symmetric and left symmetric operators on a Hilbert space. The utility of these notions are illustrated by extending some of the previous results obtained by various authors in the setting of Hilbert spaces.

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Jeet Sen, Debmalya Sain, Kallol Paul. 2021-04-17. On approximate orthogonality and symmetry of operators in semi-Hilbertian structure. https://doi.org/10.1016/j.bulsci.2021.102997

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