Norms and Numerical Radii Inequalities of Derivations Induced by Orthogonal Projections

Abstract

Properties of derivations that have been studied by different mathematicians include norms, numerical ranges, spectra and positivity. However, norms and numerical radii inequalities of derivations induced by orthogonal projections have not been considered. Therefore, the objectives of the study were to: investigate properties of derivations induced by or thogonal projections, determine norm estimates for derivations induced by orthogonal projections and determine numerical radii inequalities for derivations induced by orthogonal projections. The methodology of this research involved the use of known inequalities like Cauchy-Schwarz, tri angle inequality and parallelogram identity. Technically, we applied the norm, linearity and positivity of operators. We also considered operator tensor product. The results show that δP,Q is bounded, compact and positive for positive operators P and Q. Secondly, we showed that δP,Q has both lower and upper bounds and is utmost equal to the sum of norms of P and Q and also that δP,Q is Hermitian and is bounded above by its numerical radius. Finally, we gave power bounds for numerical radii of the δP,Q. The results of this study are useful in determining whether numerical radii of derivations implemented by orthogonal projections are equivalent to the norm of derivations. The results useful in commuta tor approximations and also in generation of constraints used in linear programming and optimal control.