Document Type : Research Paper

Author

Department of Mathematics, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran.

Abstract

We offer a new definition of $\varepsilon$-orthogonality in normed spaces, and we try to explain some properties of which. Also we introduce some types of $\varepsilon$-orthogonality in an arbitrary  $C^\ast$-algebra $\mathcal{A}$, as a Hilbert $C^\ast$-module over itself, and investigate some of its properties in such spaces. We state some results relating range-kernel orthogonality in $C^*$-algebras.

Keywords

Main Subjects

[1] J. Anderson, On normal derivations, Proc. Amer. Math. Soc, 38 (1973), pp. 135-140.

[2] L. Arambasic and R. Rajic, A strong version of the Birkhoff-James orthogonality in Hilbert C*-modules, Ann. Funct. Anal., 5 (2012), pp. 109-120.

[3] L. Arambasic and R. Rajic, The Birkhoff–-James orthogonality in Hilbert C*-modules, Linear Algebra Appl., 437 (2012), pp. 1913-1929.

[4] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J., 1 (1935), pp. 169-172.

[5] A. Blanco and A. Turnsek, On the converse of Anderson's theorem, Linear Alg ebra Appl., 424 (2007),pp. 384-389.

[6] J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure Appl. Math., 6 (2005), Art. 79.

[7] S.S. Dragomir, On approximation of continuous linear functionalsin normed linear spaces, An. Univ. Timisoara Ser. Stiint. Mat., 29 (1991), pp. 51-58.

[8] S.S. Dragomir, Semi-Inner Products and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2004.

[9] P.B. Duggal, Range kernel orthogonality of derivations, Linear Algebra appl., 304 (2000), pp. 103-108.

[10] D. Ilisevic and A. Turnsek, Approximately orthogonality preserving mappings on C*-modules, J. Math. Anal. Appl., 341 (2008), pp. 298-308.

[11] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 61 (1947), pp. 265-292.

[12] D. Keckic, Orthogonality of the range and the kernel of some elementary operators, Proc. Amer. Math. Soc., 128 (2000), pp. 3369-3377.

[13] E.C. Lance, Hilbert C*-modules. A Toolkitfor Operator Algebraists, London Mathematical Society Lecture Note Series vol. 210, Cambridge University Press, Cambridge, 1995.

[14] S. Mecheri, Non-normal derivations and orthogonality, Proc. Amer. Math. Soc. 133 (2004), no 3, 759-762.

[15] M.S. Moslehian and S.M.S. Nabavi Sales, Fuglede-Putnam type theorems via the Aluthge transform, Positivity, 17 (2013), pp. 151-162.

[16] A. Turnv sek, Generalized Anderson's inequality, J. Math. Anal. Appl., 263 (2001), pp. 121-134.