Orthogonality Of Finite Operators In Normed Spaces

Abstract

Orthogonality of operators in Hilbert spaces is a notion that has been studied for a duration of some time by many mathematicians such as Oleche, Okelo, Agure and many others. Many researchers have obtained great results concerning orthogonality of operators in normed spaces es- pecially of elementary operators but this has not been fully investigated particulary orthogonality of nite operators in normed spaces. In this study, we considered nite operators and characterized their orthogonal- ity. The objectives of this study are to: Characterize niteness of elemen- tary operators, establish orthogonality conditions for nite elementary operators and determine Birkhoff-James orthogonality for nite elemen- tary operators. The methodology involved the use of Gram Schmidt pro- cedure, Berberian Technique, Putnam Fuglede property, use of known inequalities such as Triangle inequality, Minkowski's inequality, H older's inequality, Cauchy Schwarz inequality and Bessel's inequality. We also used technical approaches such as Tensor product and Direct sum de- composition. Concerning nite elementary operators we showed that the elementary operators(Jordan elementary operator, generalized derivation, inner derivation, basic elementary operator) are nite. Then, regarding orthogonality conditions for nite elementary operators we proved that the range of nite elementary operators is orthogonal to its null space if the operators are contractive and nally on Birkhoff-James orthogonality for nite elementary operators we showed that the the range of nite ele- mentary operators is orthogonal to its kernel in terms of Birkhoff-James. The results obtained are applicable in quantum theory in estimation of the distance between the identity operator and the commutators.