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Article

Exact and Approximate Operator Parallelism

  • Ali Zamani (a1) and Mohammad Sal Moslehian (a1)
Abstract

Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra B(ℋ) of bounded linear operators acting on a Hilbert space H . Among other things, we investigate the relationship between the approximate parallelism and norm of inner derivations on B(ℋ). We also characterize the parallel elements of a C*-algebra by using states. Finally we utilize the linking algebra to give some equivalent assertions regarding parallel elements in a Hilbert C*-module.

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COPYRIGHT: © Canadian Mathematical Society 2015
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[1] Arambašić, Lj. and Rajić, R., The Birkhoff–James orthogonality in Hilbert C*-modules. Linear Algebra Appl. 437 (2012), 1913–1929. http://dx.doi.org/10.1016/j.laa.2012.05.011
[2] Arambašić, Lj. and Rajić, R., On some norm equalities in pre-Hilbert C*-modules. Linear Algebra Appl. 414 (2006), 19–28. http://dx.doi.org/10.1016/j.laa.2005.09.006
[3] Bhatia, R. and Šemrl, P., Orthogonality of matrices and some distance problems. Linear Algebra Appl. 287 (1999), 77–85. http://dx.doi.org/10.1016/S0024-3795(98)10134-9
[4] Chmielinśki, J., On an “-Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6 (2005), Art. 79, 7 pages (electronic).
[5] Dixmier, J., C*-Algebras. North-Holland, Amsterdam, 1981.
[6] Dragomir, S. S., On approximation of continuous linear functionals in normed linear spaces. An. Univ. Timis¸oara Ser. S¸ tiint¸. Mat. 29 (1991), 51–58.
[7] Fujii, J. I., Fujii, M., Moslehian, M. S., and Y. Seo, Cauchy–Schwarz inequality in semi-inner product C*-modules via polar decomposition. J. Math. Anal. Appl. 394 (2012), 835–840. http://dx.doi.org/10.1016/j.jmaa.2012.04.083
[8] Fujii, M. and Nakamoto, R., An estimation of the transcendental radius of an operator. Math. Japon. 27 (1982), 637–638.
[9] Gustafson, K. E. and Rao, D. K. M., Numerical Range. The Field of Values of Linear Operators and Matrices. Universitext, Springer-Verlag, New York, 1997.
[10] Ilišević, D. and Turnšek, A., Approximately orthogonality preserving mappings on C*-modules. J.Math. Anal. Appl. 341 (2008), 298–308. http://dx.doi.org/10.1016/j.jmaa.2007.10.028
[11] Lance, E. C., Hilbert C*-modules. A Toolkit for Operator Algebraists. London Math. Soc. Lecture Note Ser. 210, Cambridge University Press, Cambridge, 1995.
[12] Manuilov, V. M. and Troitsky, E. V., Hilbert C*-modules. In: Translations of Mathematical Monographs 226, Amer. Math. Soc., Providence, RI, 2005.
[13] McCarthy, C. A., cp. Israel J. Math. 5 (1967), 249–271. http://dx.doi.org/10.1007/BF02771613
[14] Moslehian, M. S., An operator extension of the parallelogram law and related norm inequalities. Math. Inequal. Appl. 73 (2010), 3821–3831.
[15] Murphy, G. J., C*-Algebras and Operator Theory. Academic Press, New York, 1990.
[16] Raeburn, I. and Williams, D. P., Morita equivalence and continuous-trace C*-algebras. Math. Surveys Monogr. 60, American Mathematical Society, Providence, RI, 1998.
[17] Rieffel, M. A., Standard deviation is a strongly Leibniz seminorm. arxiv:1208.4072v2.
[18] Seddik, A., On the injective norm of ni =1Ai Bi and characterization of normaloid operators. Oper. Matrices 2 (2008), 67–77. http://dx.doi.org/10.7153/oam-02-04
[19] Seddik, A., Rank one operators and norm of elementary operators. Linear Algebra Appl. 424 (2007), 177–183. http://dx.doi.org/10.1016/j.laa.2006.10.003
[20] Werner, D., An elementary approach to the Daugavet equation. In: Interaction between functional analysis, harmonic analysis, and probability (Columbia, MO, 1994, Lecture Notes in Pure and Appl. Math. 175, Dekker, New York, 1996, 449–454.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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