Mappings preserving approximate orthogonality in Hilbert $C^*$-modules
DOI:
https://doi.org/10.7146/math.scand.a-102945Abstract
We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(\delta , \varepsilon )$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(\delta , \varepsilon )$-orthogonality preserving. In particular, if $\mathscr {E}$ is a full Hilbert $\mathscr {A}$-module with $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T, S\colon \mathscr {E}\to \mathscr {E}$ are two linear mappings satisfying $|\langle Sx, Sy\rangle | = \|S\|^2|\langle x, y\rangle |$ for all $x, y\in \mathscr {E}$ and $\|T - S\| \leq \theta \|S\|$, then we show that $T$ is a $(\delta , \varepsilon )$-orthogonality preserving mapping. We also prove whenever $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T\colon \mathscr {E} \to \mathscr {F}$ is a nonzero $\mathscr {A}$-linear $(\delta , \varepsilon )$-orthogonality preserving mapping between $\mathscr {A}$-modules, then \[ \bigl \|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle \bigr \|\leq \frac {4(\varepsilon - \delta )}{(1 - \delta )(1 + \varepsilon )} \|Tx\|\|Ty\|\qquad (x, y\in \mathscr {E}). \] As a result, we present some characterizations of the orthogonality preserving mappings.
References
Arambašić, L. and Rajić, R., A strong version of the Birkhoff-James orthogonality in Hilbert $C^*$-modules, Ann. Funct. Anal. 5 (2014), no. 1, 109–120. https://doi.org/10.15352/afa/1391614575
Bakić, D. and Guljaš, B., Hilbert $C^*$-modules over $C^*$-algebras of compact operators, Acta Sci. Math. (Szeged) 68 (2002), no. 1-2, 249–269.
Bakić, D. and Guljaš, B., Wigner's theorem in a class of Hilbert $C^*$-modules, J. Math. Phys. 44 (2003), no. 5, 2186–2191. https://doi.org/10.1063/1.1556553
Chmieliński, J., Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl. 304 (2005), no. 1, 158–169. https://doi.org/10.1016/j.jmaa.2004.09.011
Chmieliński, J., Orthogonality equation with two unknown functions, Aequationes Math. 90 (2016), no. 1, 11–23. https://doi.org/10.1007/s00010-015-0359-x
Chmieliński, J., Łukasik, R., and Wójcik, P., On the stability of the orthogonality equation and the orthogonality-preserving property with two unknown functions, Banach J. Math. Anal. 10 (2016), no. 4, 828–847. https://doi.org/10.1215/17358787-3649656
Chmieliński, J. and Wójcik, P., Isosceles-orthogonality preserving property and its stability, Nonlinear Anal. 72 (2010), no. 3-4, 1445–1453. https://doi.org/10.1016/j.na.2009.08.028
Frank, M., Mishchenko, A. S., and Pavlov, A. A., Orthogonality-preserving, $C^*$-conformal and conformal module mappings on Hilbert $C^*$-modules, J. Funct. Anal. 260 (2011), no. 2, 327–339. https://doi.org/10.1016/j.jfa.2010.10.009
Ilišević, D. and Turnšek, A., Approximately orthogonality preserving mappings on $C^*$-modules, J. Math. Anal. Appl. 341 (2008), no. 1, 298–308. https://doi.org/10.1016/j.jmaa.2007.10.028
Kong, L. and Cao, H. X., Stability of orthogonality preserving mappings and the orthogonality equation, J. Shaanxi Normal Univ. Nat. Sci. Ed. 36 (2008), no. 5, 10–14.
Leung, C.-W., Ng, C.-K., and Wong, N.-C., Linear orthogonality preservers of Hilbert $C^*$-modules over $C^*$-algebras with real rank zero, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3151–3160. https://doi.org/10.1090/S0002-9939-2012-11260-2
Łukasik, R. and Wójcik, P., Decomposition of two functions in the orthogonality equation, Aequationes Math. 90 (2016), no. 3, 495–499. https://doi.org/10.1007/s00010-015-0385-8
Manuilov, V. M. and Troitsky, E. V., Hilbert $C^*$-modules, Translations of Mathematical Monographs, vol. 226, American Mathematical Society, Providence, RI, 2005.
Mojškerc, B. and Turnšek, A., Mappings approximately preserving orthogonality in normed spaces, Nonlinear Anal. 73 (2010), no. 12, 3821–3831. https://doi.org/10.1016/j.na.2010.08.007
Pambuccian, V., A logical look at characterizations of geometric transformations under mild hypotheses, Indag. Math. (N.S.) 11 (2000), no. 3, 453–462. https://doi.org/10.1016/S0019-3577(00)80009-9
Turnšek, A., On mappings approximately preserving orthogonality, J. Math. Anal. Appl. 336 (2007), no. 1, 625–631. https://doi.org/10.1016/j.jmaa.2007.03.016
Wójcik, P., On certain basis connected with operator and its applications, J. Math. Anal. Appl. 423 (2015), no. 2, 1320–1329. https://doi.org/10.1016/j.jmaa.2014.10.075
Zamani, A. and Moslehian, M. S., Approximate Roberts orthogonality, Aequationes Math. 89 (2015), no. 3, 529–541. https://doi.org/10.1007/s00010-013-0233-7
Zamani, A., Moslehian, M. S., and Frank, M., Angle preserving mappings, Z. Anal. Anwend. 34 (2015), no. 4, 485–500. https://doi.org/10.4171/ZAA/1551
Zhang, Y., Chen, Y., Hadwin, D., and Kong, L., AOP mappings and the distance to the scalar multiples of isometries, J. Math. Anal. Appl. 431 (2015), no. 2, 1275–1284. https://doi.org/10.1016/j.jmaa.2015.05.031
