A complete characterization of Birkhoff-James orthogonality in infinite dimensional normed space

In this paper, we study Birkhoff-James orthogonality of bounded linear operators and give a complete characterization of Birkhoff-James orthogonality of bounded linear operators on infinite dimensional real normed linear spaces. As an application of the results obtained, we prove a simple but useful characterization of Birkhoff-James orthogonality of bounded linear functionals defined on a real normed linear space, provided the dual space is strictly convex. We also provide separate necessary and sufficient conditions for smoothness of bounded linear operators on infinite dimensional normed linear spaces

On some geometric properties of operator spaces

In this paper we study some geometric properties like parallelism, orthogonality and semi-rotundity 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 semi-rotund points (semi-rotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semi-rotund operators and prove that $\mathbb{B}(\mathbb{X},\mathbb{Y})$ is a semi-rotund space which is not strictly convex, if $\mathbb{X},\mathbb{Y}$ are finite-dimensional Banach spaces and $\mathbb{Y}$ is strictly convex.Comment: 17 page