On $A$-parallelism and $A$-Birkhoff-James orthogonality of operators

TL;DR

This paper explores the properties of $A$-parallelism and $A$-Birkhoff-James orthogonality of bounded linear operators on Hilbert spaces, providing new characterizations, relationships, and explicit formulas related to these concepts.

Contribution

It introduces novel characterizations of $A$-parallelism and orthogonality, and relates these to the $A$-Daugavet equation and distance formulas for $A$-bounded operators.

Findings

Characterized $A$-parallelism of operators.

Established relationship between $A$-orthogonality and distance formulas.

Derived explicit formulas for the center mass of $A$-bounded operators.

Abstract

In this paper, we establish several characterizations of the $A$-parallelism of bounded linear operators with respect to the seminorm induced by a positive operator $A$ acting on a complex Hilbert space. Among other things, we investigate the relationship between $A$-seminorm-parallelism and $A$-Birkhoff-James orthogonality of $A$-bounded operators. In particular, we characterize $A$-bounded operators which satisfy the $A$-Daugavet equation. In addition, we relate the $A$-Birkhoff-James orthogonality of operators and distance formulas and we give an explicit formula of the center mass for $A$-bounded operators. Some other related results are also discussed.