Orthogonality of Finite Rank Generalized Derivations - Kaunda, Matoke Forest Collins (JOOUST, 2020)
Properties of derivations have been studied by various mathematicians for instance, norm, numerical range, spectrum, positivity among others. Range-kernel orthogonality is also a property for derivations implemented by normal operators which has been considered. However, orthogonality of derivations implemented by finite rank hyponorrnal operators has not been considered. Let A, B E FH(H) be hyponormal operators, then for all X E FH(H) it is interesting to establish the necessary and sufficient conditions for the inequality IIT + 6A,B(X) II 2: IITII to hold. The in¬equality holds for normal operators under certain conditions but these conditions arc not known for finite rank hyponorrnal operators. Thus the objectives of this study are to establish the necessary and sufficient conditions for orthogonality of the range and kernel of finite rank inner derivations implemented by hyponormal operators; establish the neces¬sary and sufficient conditions for orthogonality of the range and kernel of finite rank generalized derivations implemented by hyponormal operators and also to determine orthogonality of finite rank derivations implemented by hyponormal operators. Using Anderson's inequality for normal opera¬tors, polar decomposition and known properties of operators we have es¬tablished commutativity, contractiveness, unitarily, nilpotency, similarity and isometric properties of A1 B E FH(H) as the necessary and sufficient range-kernel orthogonality conditions for finite rank derivations imple-mented by hyponormal operators. Studies on derivations are of great use in Quantum Mechanics and Quantum Chemical Systems.