​​​​​​​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: bnyaare@yahoo.com

​Abstract

​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.

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International Journal of Modern Science and Technology

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ISSN 2456-0235