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On some geometric properties of operator spaces. (English) Zbl 1416.46022

Summary: In this article, we study some geometric properties like parallelism, orthogonality, and semirotundity in the space of bounded linear operators. We completely characterize parallelism of two compact linear operators between normed linear spaces \(\mathbb{X}\) and \(\mathbb{Y}\), assuming \(\mathbb{X}\) to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces \(\mathbb{X}\) and \(\mathbb{Y}\). We investigate parallelism and approximate parallelism in the space of bounded linear operators defined on a Hilbert space. Using the characterization of operator parallelism, we study Birkhoff-James orthogonality in the space of compact linear operators as well as bounded linear operators. Finally, we introduce the concept of semirotund points (semirotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semirotund operators and prove that \(\mathbb{B}(\mathbb{X},\mathbb{Y})\) is a semirotund space which is not strictly convex if \(\mathbb{X},\mathbb{Y}\) are finite-dimensional Banach spaces and \(\mathbb{Y}\) is strictly convex.

MSC:

46B20 Geometry and structure of normed linear spaces
46B28 Spaces of operators; tensor products; approximation properties
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References:

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