In the present paper, the generalization of the Carlson orthogonality for functionals to operators in Banach spaces has been studied. We will also investigate various properties related to the Carlsson, Birkhoff-James, and Pythagorean orthogonality for operators. Kikianty and Dragomir (2010) mentioned in their paper by stating that Pythagorean and isosceles orthogonality through the medium of 2 − HH norm satisfies the non-degeneracy, symmetry and continuity properties without mentioning detailed proof. This paper provides the complete proof of these properties as well as the equivalency of additivity and homogeneity of the isosceles orthogonality with the help of 2 − HH norm. In the case of norm attaining bounded linear operators in a Hilbert space, we prove an equivalence relation between the Pythagorean and isosceles orthogonality including result that the Pythagorean implies Birkhoff-James, whereas the converse is not necessarily true. Furthermore, we will study a new particular case of Carlsson orthogonality and show that this orthogonality implies Birkhoff-James orthogonality, but the converse may not be true.
Ojha, Bhuwan P. and Bajracharya, Prakash M.
Orthogonality in Terms of 2-HH Norm and Bounded Linear Operators in Banach Spaces,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 16,
1, Article 35.
Available at: https://digitalcommons.pvamu.edu/aam/vol16/iss1/35