On Properties of Hilbert Space Operators and Applications

Abstract

A lot of studies in operator theory are useful in applications to other disciplines like engineering and medical sciences among others. One such study is characterization

of properties of operators on Hilbert spaces. A lot of useful results have been

obtained on norms of normal operators. However, characterization of normality

and norm-attainability of these operators have not been exhausted. We outline the

theory of normal, self-adjoint and norm-attainable operators. This study sought

to investigate the conditions for norm-attainability of self-adjoint operators; and

orthogonality of self-adjoint norm-attainable normal operators. The methodology

involved use of inner products, tensor products and some known mathematical

inequalities like Cauchy-Schwarz inequality, parallelogram identity and the

triangle inequality. Results showed a strong relationship between normal operators

and norm-attainable operators’ i.e. normal operators are norm-attainable if they

are self-adjoint. These results are useful in generating quantum bits and estimation

of ground state energies of various molecules like ethane in quantum theory