Mappings preserving approximate orthogonality in Hilbert C^*-modules - Moslehian, Mohammad Sal; Zamani, Ali (MATHEMATICA SCANDINAVICA, 2018)
We introduce a notion of approximate orthogonality preserving mappings between Hilbert C^*-modules. We define the concept of (δ , ε )-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be (δ , ε )-orthogonality preserving. In particular, if E is a full Hilbert A-module with K(H)A B(H) and T, SE→ E are two linear mappings satisfying |Sx, Sy| = \|S\|^2|x, y| for all x, yE and \|T - S\| ≤ θ \|S\|, then we show that T is a (δ , ε )-orthogonality preserving mapping. We also prove whenever K(H)A B(H) and TE → F is a nonzero A-linear (δ , ε )-orthogonality preserving mapping between A-modules, then \[ \|Tx, Ty- \|T\|^2x, y\|≤ 4(ε - δ )(1 - δ )(1 + ε ) \|Tx\|\|Ty\|(x, yE). \] As a result, we present some characterizations of the orthogonality preserving mappings.