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 and , assuming to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces and . 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 is a semirotund space which is not strictly convex if are finite-dimensional Banach spaces and is strictly convex.
"On some geometric properties of operator spaces." Banach J. Math. Anal. 13 (1) 174 - 191, January 2019. https://doi.org/10.1215/17358787-2018-0021