On Numerical Ranges And Spectra Of Posinormal Operators

Abstract

Hilbert space operators have been studied by many mathematicians. These operators are of great importance since they are useful in formulation of principles of mathematical analysis and quantum mechanics. The operators include normal operators, posinormal operators, hyponormal operators, normaloid operators among others. Certain properties of posinormal operators have been characterized like continuity and linearity but numerical ranges and spectra of posinormal operators have not been considered. Also the relationship between the numerical range and spectrum has not been determined for posinormal operators. The objectives of this study have been: to investigate numerical ranges of posinormal operators, to investigate the spectra of posinormal operators and to establish the relationship between the numerical range and spectrum of a posinormal operator. The methodology involved use of known inequalities like Cauchy- Schwartz inequality and the polarization identity to determine the numerical range and spectrum of posinormal operators and our technical approach involved use of tensor products. We have shown that the numerical range of a posinormal operator A is nonempty, contains zero and is an ellipse whose foci are the eigenvalues of A. We have also proved that the spectrum of a bounded posinormal operator A acting on a complex Hilbert space H satisfies Xia’s property; and doubly commuting n-tuples of posinormal operators are jointly normaloid. The results obtained are applicable in classification of Hilbert space operators and shall be applied in other fields like quantum information theory to optimize minimal output entropy of quantum channel; to detect entanglement using positive maps; and for local distinguishability of unitary operators.