Arpita Mal is an Inspire Faculty Fellow at the Department of Mathematics, Indian Institute of Science, Bengaluru, under the mentorship of Prof. Apoorva Khare. Dr. Mal completed one year of postdoctoral research with SERB National Post-Doctoral fellowship at the same department, under the same mentor. She was also awarded an NBHM postdoctoral fellowship. She completed her Ph.D. at Jadavpur University, Kolkata, under the supervision of Prof. Kallol Paul in 2021. Dr. Mal is Recipient of a Gold Medal for securing the first-class-first position at M.Sc. from Jadavpur University. With 25 research articles in different international journals of high reputation, her research interest includes the geometry of Banach space and operator space. Kallol Paul is a Professor at the Department of Mathematics, Jadavpur University, Kolkata. He completed his B.Sc. in Mathematics from Gauhati University, Assam, and M.Sc. from Jadavpur University, securing the first-class-first position in both the examinations. His broad area of research is functional analysis and operator theory. With more than 26 years of teaching and research experience, he has so far guided 16 students for the successful completion of their Ph.D. degrees, and another 6 students are presently pursuing their doctoral research under his supervision. He has published more than 140 papers in various international journals of repute. His current area of teaching includes real analysis, metric spaces, and operator theory. Presently, he is actively involved in research on a variety of topics, including numerical radius inequalities, Birkhoff–James orthogonality, and its applications in the study of Banach spaces. With a vast international network of collaborators, Prof. Paul has played a crucial role in establishing Jadavpur University as an important center of research activity on Banach space theory.Debmalya Sain is an Assistant Professor at the Department of Mathematics and Computing, Indian Institute of Information Technology Raichur, Karnataka, India. An alumnus of Ramakrishna Mission Vidyapith, Purulia, he completed his Ph.D. degree in 2015 under the guidance of Prof. Kallol Paul. A recipient of multiple Gold Medals in his undergraduate and postgraduate studies, Dr. Sain stood first-class-first in both his B.Sc. and M.Sc. examinations at Jadavpur University, Kolkata. He completed multiple post-doctoral fellowships at the Indian Institute of Science, Bengaluru, under the mentorship of Prof. Gadadhar Misra and Prof. Apoorva Khare. He has also received many international grants and awards, including the prestigious Maria Zambrano grant to pursue his research at the Universidad de Granada, Spain, under the guidance of Prof. Miguel Martin. Dr. Sain has authored more than 75 articles in various reputed international journals. Apart from acting as a referee and reviewer for many international journals, he is an invited member of the editorial board of the Advances in Operator Theory journal, published by Birkhauser.

“This interesting book deals with various ramifications of the geometric notion of Birkhoff-James orthogonality in both concrete and abstract situations. … this well-written monograph is a must-read for anyone interested in aspects of Birkhoff-James orthogonality and its interplay with the geometry of Banach spaces.” (T.S.S.R.K. Rao, zbMATH 1547.46002, 2024)

• 1 Notations and Terminologies.- 2 Basic theory of B-J orthogonality in Banach space.- 3 Operator norm attainment.- 4 B-J orthogonality of operators.- 5 Approximate B-J orthogonality.- 6 Smoothness and k−smoothness of operators.- 7 Symmetry of the B-J orthogonality.- 8 Extreme contractions.

Arpita Mal is an Inspire Faculty Fellow at the Department of Mathematics, Indian Institute of Science, Bengaluru, under the mentorship of Prof. Apoorva Khare. Dr. Mal completed one year of postdoctoral research with SERB National Post-Doctoral fellowship at the same department, under the same mentor. She was also awarded an NBHM postdoctoral fellowship. She completed her Ph.D. at Jadavpur University, Kolkata, under the supervision of Prof. Kallol Paul in 2021. Dr. Mal is Recipient of a Gold Medal for securing the first-class-first position at M.Sc. from Jadavpur University. With 25 research articles in different international journals of high reputation, her research interest includes the geometry of Banach space and operator space. Kallol Paul is a Professor at the Department of Mathematics, Jadavpur University, Kolkata. He completed his B.Sc. in Mathematics from Gauhati University, Assam, and M.Sc. from Jadavpur University, securing the first-class-first position in both the examinations. His broad area of research is functional analysis and operator theory. With more than 26 years of teaching and research experience, he has so far guided 16 students for the successful completion of their Ph.D. degrees, and another 6 students are presently pursuing their doctoral research under his supervision. He has published more than 140 papers in various international journals of repute. His current area of teaching includes real analysis, metric spaces, and operator theory. Presently, he is actively involved in research on a variety of topics, including numerical radius inequalities, Birkhoff–James orthogonality, and its applications in the study of Banach spaces. With a vast international network of collaborators, Prof. Paul has played a crucial role in establishing Jadavpur University as an important center of research activity on Banach space theory.Debmalya Sain is an Assistant Professor at the Department of Mathematics and Computing, Indian Institute of Information Technology Raichur, Karnataka, India. An alumnus of Ramakrishna Mission Vidyapith, Purulia, he completed his Ph.D. degree in 2015 under the guidance of Prof. Kallol Paul. A recipient of multiple Gold Medals in his undergraduate and postgraduate studies, Dr. Sain stood first-class-first in both his B.Sc. and M.Sc. examinations at Jadavpur University, Kolkata. He completed multiple post-doctoral fellowships at the Indian Institute of Science, Bengaluru, under the mentorship of Prof. Gadadhar Misra and Prof. Apoorva Khare. He has also received many international grants and awards, including the prestigious Maria Zambrano grant to pursue his research at the Universidad de Granada, Spain, under the guidance of Prof. Miguel Martin. Dr. Sain has authored more than 75 articles in various reputed international journals. Apart from acting as a referee and reviewer for many international journals, he is an invited member of the editorial board of the Advances in Operator Theory journal, published by Birkhauser.

“This interesting book deals with various ramifications of the geometric notion of Birkhoff-James orthogonality in both concrete and abstract situations. … this well-written monograph is a must-read for anyone interested in aspects of Birkhoff-James orthogonality and its interplay with the geometry of Banach spaces.” (T.S.S.R.K. Rao, zbMATH 1547.46002, 2024)

• 1 Notations and Terminologies.- 2 Basic theory of B-J orthogonality in Banach space.- 3 Operator norm attainment.- 4 B-J orthogonality of operators.- 5 Approximate B-J orthogonality.- 6 Smoothness and k−smoothness of operators.- 7 Symmetry of the B-J orthogonality.- 8 Extreme contractions.