Taiwanese Journal of Mathematics

On Orthogonality of Elementary Operators in Norm-attainable Classes

Nyaare Benard Okelo

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Various notions of orthogonality of elementary operators have been characterized by many mathematicians in different classes. In this paper, we characterize orthogonality of these operators in norm-attainable classes. We first give necessary and sufficient conditions for norm-attainability of Hilbert space operators then we give results on orthogonality of the range and the kernel of elementary operators when they are implemented by norm-attainable operators in norm-attainable classes.

Article information

Taiwanese J. Math., Advance publication (2019), 12 pages.

First available in Project Euclid: 7 May 2019

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Digital Object Identifier

Primary: 47B47: Commutators, derivations, elementary operators, etc.
Secondary: 47A30: Norms (inequalities, more than one norm, etc.)

norm-attainable class range-kernel orthogonality elementary operator


Okelo, Nyaare Benard. On Orthogonality of Elementary Operators in Norm-attainable Classes. Taiwanese J. Math., advance publication, 7 May 2019. doi:10.11650/tjm/190502. https://projecteuclid.org/euclid.twjm/1557194426

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  • J. Anderson, On normal derivations, Proc. Amer. Math. Soc. 38 (1973), 135–140.
  • A. Bachir and A. Segres, Numerical range and orthogonality in normed spaces, Filmat 23 (2009), no. 1, 21–41.
  • G. Bachman and L. Narici, Functional Analysis, Dover Publications, Mineola, NY, 2000.
  • C. Benitez, Orthogonality in normed linear spaces: a classification of the different concepts and some open problems, Rev. Mat. Univ. Complut. Madrid 2 (1989), suppl., 53–57.
  • R. Bhatia and P. Šemrl, Orthogonality of matrices and some distance problems, Linear Algebra Appl. 287 (1999), no. 1-3, 77–85.
  • D. K. Bhattacharya and A. K. Maity, Semilinear tensor product of $\Gamma$-Banach algebras, Ganita 40 (1989), no. 1-2, 75–80.
  • F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, New York, 1973.
  • J. O. Bonyo and J. O. Agure, Norm of a derivation and hyponormal operators, Int. J. Math. Anal. (Ruse) 4 (2010), no. 13-16, 687–693.
  • S. Bouali and Y. Bouhafsi, On the range of the elementary operator $X \mapsto AXA-X$, Math. Proc. R. Ir. Acad. 108 (2008), no. 1, 1–6.
  • F. M. Brückler, Tensor products of $C^*$-algebras, operator spaces and Hilbert $C^*$-modules, Math. Commun. 4 (1999), no. 2, 257–268.
  • M. Cabrera García and Á. Rodríguez Palacios, Non-degenerately ultraprime Jordan-Banach algebras: a Zel'manovian treatment, Proc. London Math. Soc. (3) 69 (1994), no. 3, 576–604.
  • J. A. Canavati, S. V. Djordjević and B. P. Duggal, On the range closure of an elementary operator, Bull Korean Math. Soc. 43 (2006), no. 4, 671–677.
  • J. Chmieliński, T. Stypuła and P. Wójcik, Approximate orthogonality in normed spaces and its applications, Linear Algebra Appl. 531 (2017), 305–317.
  • H.-K. Du, Y.-Q. Wang and G.-B. Gao, Norms of elementary operators, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1337–1348.
  • T. K. Dutta, H. K. Nath and R. C. Kalita, $\alpha$-derivations and their norm in projective tensor products of $\Gamma$-Banach algebras, Internat. J. Math. Math. Soc. 21 (1998), no. 2, 359–368.
  • P. Gajendragadkar, Norm of a derivation on a von Neumann algebra, Trans. Amer. Math. Soc. 170 (1972), 165–170.
  • G. X. Ji and H. K. Du, Norm attainability of elementary operators and derivations, Northeast. Math. J. 10 (1994), no. 3, 396–400.
  • R. V. Kadison, E. C. Lance and J. R. Ringrose, Derivations and automorphisms of operator algebra II, J. Functional Analysis 1 (1967), 204–221.
  • D. J. Kečkić, Orthogonality of the range and the kernel of some elementary operators, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3369–3377.
  • ––––, Orthogonality in $\mathfrak{S}_{1}$ and $\mathfrak{S}_{\infty}$ spaces and normal derivations, J. Operator Theory 51 (2004), no. 1, 89–104.
  • F. Kittaneh, Normal derivations in norm ideals, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1779–1785.
  • E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, 1978.
  • J. Kyle, Norms of derivations, J. London Math. Soc. (2) 16 (1997), no. 2, 297–312.
  • B. Magajna, The norm of a symmetric elementary operator, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1747–1754.
  • M. Mathieu, Properties of the product of two derivations of a $C^*$-algebra, Canad. Math. Bull. 32 (1989), no. 4, 490–497.
  • ––––, More properties of the product of two derivations of a $C^*$-algebra, Bull. Austral. Math. Soc. 42 (1990), no. 1, 115–120.
  • ––––, Elementary operators on Calkin algebras, Irish Math. Soc. Bull. 46 (2001), 33–42.
  • S. Mecheri, Finite operators, Demonstratio Math. 35 (2002), no. 2, 357–366.
  • ––––, On the range and the kernel of the elementary operators $\sum_{i=1}^{n} A_{i} XB_{i} - X$, Acta Math. Univ. Comenian. (N.S.) 72 (2003), no. 2, 191–196.
  • G. J. Murphy, $C^*$-algebras and Operator Theory, Academic Press, Boston, MA, 1990.
  • N. B. Okelo, J. O. Agure and D. O. Ambogo, Norms of elementary operators and characterization of norm-attainable operators, Int. J. Math. Anal. (Ruse) 4 (2010), no. 21-24, 1197–1204.
  • N. B. Okelo, J. O. Agure and P.O. Oleche, Various notions of orthogonality in normed spaces, Acta Math. Sci. Ser. B (Engl. Ed.) 33 (2013), no. 5, 1387–1397.
  • K. Paul, D. Sain and A. Mal, Approximate Birkhoff-James orthogonality in the space of bounded linear operators, Linear Algebra Appl. 537 (2018), 348–357.
  • A. Turnšek, Orthogonality in $\mathscr{C}_{p}$ classes, Monatsh. Math. 132 (2001), no. 4, 349–354.
  • J. P. Williams, Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129–136.
  • P. Wójcik, The Birkhoff orthogonality in pre-Hilbert $C^*$-modules, Oper. Matrices 10 (2016), no. 3, 713–729.
  • ––––, Orthogonality of compact operators, Expo. Math. 35 (2017), no. 1, 86–94.
  • ––––, Birkhoff orthogonality in classical $M$-ideals, J. Aust. Math. Soc. 103 (2017), no. 2, 279–288.