Birkhoff–James orthogonality is one of the most fundamental and widely studied notions of orthogonality in Banach spaces, serving as a natural extension of the inner-product induced orthogonality in Hilbert spaces. In this talk, we explore the geometric and analytical features of Birkhoff–James orthogonality and highlight its role in understanding the geometry of Banach space. We study this notion of orthogonality in the space of all bounded linear operators defined on Hilbert spaces. In particular, we discuss various symmetry properties associated with Birkhoff–James orthogonality and their connections with surjective isometries.
We also study its applications to best approximation problem. Special emphasis is placed on the Fermat–Torricelli problem, a well celebrated classical problem in location theory.
Dr. Shamim Sohel is currently an ANRF National Postdoctoral Fellow in the Department of Mathematics at IIT Madras, under the guidance of Dr. Surjit Kumar. He completed his PhD at Jadavpur University, Kolkata.
His research interests broadly lie in functional analysis and the geometry of Banach spaces. In particular, his work focuses on best approximation and best coapproximation, as well as geometric notions such as extreme points and smooth points in Banach space theory, with Birkhoff–James orthogonality serving as a central tool.
