Some study of the approximate Birkhoff orthogonality and orthogonality of bounded linear operators

In this note, we are mainly interested in orthogonality, including Birkhoff orthogonality, isosceles orthogonality, Pythagorean orthogonality and approximate Birkhoff orthogonality perpendicular to(B epsilon). First, we show the relation between linear functionals and hyperplanes by means of approximate Birkhoff orthogonality. Second, we establish the sufficient conditions for Birkhoff orthogonality and approximate Birkhoff orthogonality. Then some sufficient and necessary conditions for isosceles orthogonality and Pythagorean orthogonality of bounded operators are shown. In addition, we characterize the inner product space in terms of functional orthogonality.