International Journal of Modern Science and Technology
January 2018, Vol. 3, No 1, pp 6-9.
Characterization of Inner Derivations induced by Norm-attainable Operators
M. O. Oyake, N. B. Okelo, O. Ongati
School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology,
P.O. Box 210-40601, Bondo-Kenya.
*Corresponding author’s e-mail: firstname.lastname@example.org
In the present paper, results on characterization of inner derivations in Banach algebras are discussed. Some techniques are employed for derivations due to Mecheri, Hacene, Bounkhel and Anderson. Let H be an infinite dimensional complex Hilbert space and B(H) the algebra of all bounded linear operators on H. A generalized derivation δ: B(H) → B(H) is defined by δA,B(X) = AX −XB, for all X ∈ B(H) and A,B fixed in B(H). An inner derivation is defined by δA(X) = AX −XA, for all X ∈ B(H) and A fixed in B(H). Norms of inner derivations have been investigated by several mathematicians. However, it is noted that norms of inner derivations implemented by norm-attainable operators have not been considered to a great extent. In this study, we investigate properties of inner derivations which are strictly implemented by norm-attainable and we determine their norms. The derivations in this work are all implemented by norm-attainable operators. The results show that these derivations admit tensor norms of operators.
Keywords: Banach space; Hilbert space; Inner Derivation; Norms; Tensor Products.
- Anderson A. On normal derivations. Proc Amer Math Soc. 1979;38:129-140.
- Bounkhel M. On Minimizing the norm of Linear maps in Cp-classes. Applied Sciences. 2006;8:40-47.
- Charalambos Aliprantis D, Owen B. Principles of Real Analysis Academic Press, 3rd Edition, 1998.
- Christopher H. Banach and Hilbert Space Review. Lecture Notes. 2006;1-13.
- Daryoush B, Encyeh DN. Introduction of Frechet and Gateaux Derivative. Applied Mathematical Sciences. 2008;2:975-980.
- Keckic D. Orthogonality in and C1-spaces and Cα normal derivations. J Operator Theory. 2004;51:89-104.
- Kittaneh F. Normal derivations in normal ideals. Proc Amer Math Soc. 1995;123:1779-1785.
- Mecheri S. Some recent results on operator commutators and related operators with applications. Lecture Monograph. 2010.
- Mecheri S, Hacene M. Gateaux Derivative and Orthogonality in Cα, Journal of Inequalities in Pure and Applied Mathematics. 2012;20:275-284.
- Mecheri S, Bounkhel M. Global minimum and Orthogonality in C1-classes. J Math Anal Appl. 2003;1:51-60.
- Mecheri S. The Gateaux Derivative Orthogonality in Cα, Lecture Notes. 1991.
- Vijayabalaji S, Shyamsundar G. Interval-valued intuitionistic fuzzy transition matrices. International Journal of Modern Science and Technology. 2016;1(2):47-51.
- Judith JO, Okelo NB, Roy K, Onyango T. Numerical Solutions of Mathematical Model on Effects of Biological Control on Cereal Aphid Population Dynamics. International Journal of Modern Science and Technology. 2016;1(4):138-143.
- Judith JO, Okelo NB, Roy K, Onyango T. Construction and Qualitative Analysis of Mathematical Model for Biological Control on Cereal Aphid Population Dynamics. International Journal of Modern Science and Technology. 2016;1(5):150-158.
- Vijayabalaji S, Sathiyaseelan N. Interval-Valued Product Fuzzy Soft Matrices and its Application in Decision Making. International Journal of Modern Science and Technology. 2016;1(7):159-163.
- Chinnadurai V, Bharathivelan K. Cubic Ideals in Near Subtraction Semigroups. International Journal of Modern Science and Technology. 2016;1(8):276-282.