NORMS AND NORM-ATTAINABILITY OF NORMAL OPERATORS AND THEIR APPLICATIONS

Abstract

The study of norms of operators forms a very important aspect in functional analysis, operator theory and its applications to economics, quantum chemistry and quantum computing amongst other fields. A lot of results have been obtained on norms of normal operators particularly by Kitanneh, Dragomir and Stacho among others. However, characterizations of norm-attainable operators have not been exhausted. The pending question that remains unanswered is: What are the necessary and sufficient conditions for normal operators to be norm-attainable? Moreover, what are the norms of these operators if the norm-attainability suffices? Therefore, in this paper, we present norms of operators in Hilbert spaces. We outline the theory of normal, self-adjoint and norm-attainable operators. The objectives of the study are: To determine norms of normal operators; To establish conditions for norm-attainability of normal operators; and to investigate norms of selfadjoint norm-attainable operators. The methodology involved the use of inner products, tensor products and some known mathematical inequalities like Cauchy-Schwarz inequality and the triangle inequality. The results obtained show that normaloid and normal operators are norm-attainable if there exist a unit vector x in the Hilbert space which is unique such that for any operator T,T Tx . Furthermore, the operators are normattainable if they are self-adjoint. These results concur with the results of Qui Bao Gao for compact operators when the Hilbert space is taken to be infinite dimensional and complex. In conclusion, the results obtained are useful in quantum computing in generating quantum bit. In genetics the results are useful in the determination of DNA results of parents and offspring. Key words: Hilbert space, normality, norm-attainability, self-adjoint operators, tensor products