We introduce and study a generalized approximate orthogonality relation in real normed linear spaces, namely, approximate -orthogonality. We investigate the relation between this generalized approximate orthogonality and approximate Birkhoff–James orthogonality. In particular, we show that every approximately -orthogonality-preserving linear mapping is necessarily a scalar multiple of an almost isometry.
"Approximate -orthogonality and its preservation." Ann. Funct. Anal. 10 (4) 483 - 495, November 2019. https://doi.org/10.1215/20088752-2019-0004