ON THE EXPONENTIAL AND APPROXIMATE POINT SPECTRUM OF A BOUNDED LINEAR OPERATOR IN A BANACH SPACE

Abstract

The exponential and spectrum of bounded linear operators have been investigated to a fair extend with regard to their characterization. Numerous studies have emphasized on the exponential stability of semigroups of linear operators in terms of its actions and the asymptotic behaviours. Similarly, Banach spaces have been studied and various f indings have been obtained. Due to the fundamental fact that operators on Banach spaces are bounded and linear, our main objective in this study was to investigate the exponential and approximate point spectrum of a bounded linear operator in Banach spaces. Specifically we have defined the operator exp(T) for a bounded linear operator T in a Banach space X and determined if exp(T) is invertible. We have shown that if S is a bounded linear operator in X and S commutes with T then exp(S +T) equals exp(T).exp(S) . If H is a Hilbert space and T is a bounded linear operator in H which is normal such that T commutes with a bounded linear operator S in H, then S commutes with the adjoint of T. We have shown that the converse of this statement holds. The spectrum of T can be expressed as the union of the approximate point spectrum of T and the compression spectrum of T. These two sets may overlap. For A a bounded linear operator in H, we have shown that the boundary of the spectrum of A is contained in the approximate point spectrum of A. Finally we have shown that if A and B are bounded linear operators in H which are similar, then A and B have the same spectrum, point spectrum, approximate point spectrum and compression spectrum. There exist some known results in spectral theory, for example the spectral mapping theorem which we have explored and demonstrated that the exponential and approximate point spectrum are independent of the Banach space. We have used the Fuglede’s Theorem to its converse. We have also exploited the properties of the spectrum and used the convergence norm to determine our result. The results we have obtained in this study will be of great importance to the study of the theory of analysis and spectral theory