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

xIn this note, we are mainly interested in orthogonality, including Birkhoff orthogonality, isosceles orthogonality, Pythagorean orthogonality and approximate Birkhoff orthogonality perpendicular to B-is an element of. 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.

Automated summary

This note mainly focuses on orthogonality, examining properties and conditions of Birkhoff orthogonality, isosceles orthogonality, Pythagorean orthogonality, and approximate Birkhoff orthogonality perpendicular to B. The note establishes relations between linear functionals and hyperplanes through approximate Birkhoff orthogonality, and provides sufficient and necessary conditions for various types of orthogonality of bounded operators. Additionally, the note characterizes inner product spaces in terms of functional orthogonality.