Approximate ρλv-orthogonality and its preservation

Abstract

We introduce and study a generalized approximate orthogonality relation in real normed linear spaces, namely, approximate ${\rho }_{\lambda }^v$-orthogonality. We investigate the relation between this generalized approximate orthogonality and approximate Birkhoff–James orthogonality. In particular, we show that every approximately ${\rho }_{\lambda }^v$-orthogonality-preserving linear mapping is necessarily a scalar multiple of an almost isometry.