Executive Summary : | The geometry of Banach spaces differs significantly from Hilbert spaces due to the absence of an in-built inner product. This makes the study of isometric operators, extreme contractions, and k-smoothness in the setting of bounded linear operators between Banach spaces more intriguing. The study of geometric properties of operator spaces is a fertile area of research. This project focuses on the preservation of geometric properties of bounded linear operators defined between Banach spaces and their applications to the study of isometric operators between Banach spaces. Birkhoff-James orthogonality, a useful and important notion of orthogonality, has been completely characterized and approximated in the space of bounded linear operators between Banach spaces. The main motivation is to study preservation of geometric properties locally using Birkhoff-James orthogonality and approximate Birkhoff-James orthogonality in a Banach space, similar to inner product orthogonality in a Hilbert space. Extreme contractions in Banach spaces are not yet fully known, even for operators defined between finite-dimensional Banach spaces. The project aims to characterize isometric operators and extreme contractions in two-dimensional Banach spaces, n-dimensional Banach spaces, and infinite-dimensional Banach spaces. The results will have a significant impact on the study of isometric operators defined between Banach spaces and help better understand the difference between Banach space geometry and Euclidean geometry. |