Research Projects

The norm of a bounded linear operator between Banach spaces is a fundamental numeric constant associated with it. In this context, a natural query is whether the norm is actually attained at some unit vector. A standard compactness argument ensures that the answer to this question is in the affirmative in the finite-dimensional case. On the other hand, the situation is far more non-trivial in the infinite-dimensional case, where norm attaining operators may not even be dense in the space of bounded linear operators. One of the most celebrated programs in the isometric theory of Banach spaces, initiated by Lindenstrauss in the 1960s, is the study of the density properties of norm attaining operators between Banach spaces. Moreover, many important analytic and geometric properties of bounded linear operators, such as Gateaux differentiability and the property of being an extreme contraction, depend on their norm attainment sets. All these observations point to the conclusion that a complete description of the (possibly empty) norm attainment set of a bounded linear operator is crucial for understanding its geometric attributes. Furthermore, a moment's reflection reveals that the norm attainment problem is not unique to bounded linear operators. Indeed, for every member of the family of continuous functions between Banach spaces, including multilinear operators, n-homogeneous polynomials, and Lipschitz maps, the norm attainment of the concerned function is important for understanding its properties. While the norm attainment problem has been studied in considerable detail for the particular case of bounded linear operators, it has not received the due attention for the previously mentioned other mathematical operations between Banach spaces. The main goal of the proposed project is to obtain a unified treatment of the norm attainment problem for various mathematical operations, and to apply the obtained results in the geometric theory of Banach spaces. Very recently, it has been proved that a necessary condition for the norm attainment of a bounded linear operator is the local preservation of the so-called Birkhoff-James orthogonality at the concerned point, along a hyperplane to which the point is orthogonal. Applying this observation, complete descriptions of the operator norm attainment sets have been obtained in some specific cases. In light of this result, we aim to study the norm attainment of a continuous function between Banach spaces, from the perspective of local preservation of orthogonality. We are also interested in applying the principle of preservation of orthogonality in describing the isometry group of a Banach space, which is a historically important problem in Functional Analysis and Operator Theory. Another ambitious target of this project is to develop a suitable notion of orthogonality in the setting of topological vector spaces which may have important implications in the isomorphic theory of Banach spaces.