Characterizations of Operator Birkhoff–James Orthogonality

Moslehian, Mohammad Sal ; Zamani, Ali

Canadian mathematical bulletin, Tome 60 (2017) no. 4, pp. 816-829

Résumé

In this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert ${{C}^{*}}$ -modules and certain elements of $\mathbb{B}\left( H \right)$ . Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(H)$ we prove that if the norm attaining set ${{\mathbb{M}}_{T}}$ is a unit sphere of some finite dimensional subspace ${{H}_{0}}$ of $H$ and $||T|{{|}_{{{H}_{0}}\bot }}\,<\,\,||T||$ , then for every $S\in \mathbb{B}(H)$ , $T$ is the strong Birkhoff–James orthogonal to $S$ if and only if there exists a unit vector $\xi \in {{H}_{0}}$ such that $||T||\xi =\,|T|\xi $ and ${{S}^{*}}T\xi =0$ . Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product ${{C}^{*}}$ -modules.

Détail

DOI : 10.4153/CMB-2017-004-5

Mots-clés : 46L05, 46L08, 46B20, Hilbert C*-module, BirkhoÒ–James orthogonality, strong BirkhoÒ–James orthogonality, approximate orthogonality

Bibliographie

[1] [1] Aldaz, J. M., S. Barza, M. Fujii, and Moslehian, M. S., Advances in operator Cauchy-Schwarz inequalities and their reverses. Ann. Funct. Anal. 6(2015), no. 3, 275–295. Google Scholar | DOI

[2] [2] Arambasic, Lj. and R. Rajic, A strong version of the Birkhoff-–James orthogonality in Hilbert C*-modules. Ann. Funct. Anal. 5(2014), no. 1, 109–120. Google Scholar | DOI

[3] [3] Arambasic, Lj., On three concepts of orthogonality in Hilbert C* -modules. Linear Multilinear Algebra 63(2015), no. 7, 1485–1500. Google Scholar | DOI

[4] [4] Bhatia, R. and P. Semrl, Orthogonality of matrices and some distance problems. Linear Algebra Appl. 287(1999), no. 1–3, 77-85. Google Scholar | DOI

[5] [5] Bhattacharyya, T. and P. Grover, Characterization of Birkhoff-–James orthogonality. J. Math. Anal. Appl. 407(2013), no. 2, 350–358. Google Scholar | DOI

[6] [6] Birkhoff, G., Orthogonality in linear metric spaces. Duke Math. J. 1(1935), no. 2,169-172. Google Scholar | DOI

[7] [7] Chmieliiiski, J., On an e-Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6(2005), no. 3, Art. 79. Google Scholar

[8] [8] Chmieliiiski, J., Linear mappings approximately preserving orthogonality. J. Math. Anal. Appl. 304(2005), 158-169. Google Scholar | DOI

[9] [9] Chmieliiiski, J., Orthogonality equation with two unknown functions. Aequationes Math. 90(2016), no. 1, 11–23. Google Scholar | DOI

[10] [10] Diogo, C., Algebraic properties of the set of operators with 0 in the closure of the numerical range. Oper. Matrices 9(2015), no. 1, 83–93. Google Scholar | DOI

[11] [11] Dixmier, J., C* -Algebras. North-Holland, Amsterdam, 1981. Google Scholar

[12] [12] 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. Google Scholar | DOI

[13] [13] Ghosh, P., D. Sain and K. Paul, Orthogonality of bounded linear operators. Linear Algebra Appl. 500(2016), 43–51. Google Scholar | DOI

[14] [14] Grover, P., Orthogonality of matrices in the Ky Fan k-norms. Linear Multilinear Algebra 65(2017), no. 3, 496–509. Google Scholar | DOI

[15] [15] Ilisevic, D. and A. Turnsek, Approximately orthogonality preserving mappings on C* -modules. J. Math. Anal. Appl. 341(2008), no. ,1 298–308. Google Scholar | DOI

[16] [16] James, R. C., Orthogonality in normed linear spaces. Duke Math. J. 12(1945), 291–302. Google Scholar | DOI

[17] [17] Lance, E. C., Hilbert C* -modules. A toolkit for operator algebraists. London Mathematical Society Lecture Note Series, 210, Cambridge University Press, Cambridge, 1995. Google Scholar | DOI

[18] [18] Leung, C.-W., C.-K. Ng, and N.-C. Wong, Linear orthogonality preservers of Hilbert C*-modules. J. Operator Theory 71(2014), no. 2, 571–584. Google Scholar | DOI

[19] [19] Mojskerc, B. and A. Turnsek, Mappings approximately preserving orthogonality in normed spaces. Nonlinear Anal. 73(2010), no. 12, 3821–3831. Google Scholar | DOI

[20] [20] Murphy, G. J., C*-Algebras and operator theory. Academic Press, Boston, MA, 1990. Google Scholar

[21] [21] Paul, K., D. Sain, and P. Ghosh, Birkhoff-–fames orthogonality and smoothness of bounded linear operators. Linear Algebra Appl. 506(2016), 551–563. Google Scholar | DOI

[22] [22] Sain, D., K. Paul, and S. Hait, Operator norm attainment and Birkhoff-–James orthogonality. Linear Algebra Appl. 476(2015), 85–97. Google Scholar | DOI

[23] [23] Wojcik, P., Norm-parallelism in classical M-ideals. Indag. Math., to appear. Google Scholar | DOI

[24] [24] Zamani, A. and Moslehian, M. S., Exact and approximate operator parallelism. Canad. Math. Bull. 58(2015), no. 1, 207–224. Google Scholar | DOI

[25] [25] Zamani, A., Moslehian, M. S., and M. Frank, Angle preserving mappings. Z. Anal. Anwend. 34(2015), no. 4, 485–500. Google Scholar | DOI