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.
