Some Spectral Aspects of the Operator over the Sequence Spaces and

The main idea of the present paper is to compute the spectrum and the fine spectrum of the generalized difference operator over the sequence spaces . The operator denotes a triangular sequential band matrix defined by with for , where or , ; the set nonnegative integers and is either a constant or strictly decreasing sequence of positive real numbers satisfying certain conditions. Finally, we obtain the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of the operator over the sequence spaces and . These results are more general and comprehensive than the spectrum of the difference operators , , , , and and include some other special cases such as the spectrum of the operators , , and over the sequence spaces or . 1. Introduction, Preliminaries, and Definitions In analysis, operator theory is one of the important branch of mathematics which has vast applications in the field applied science and engineering. Operator theory deals with the study related to different properties of operators such as their inverse, spectrum, and fine spectrum. Since the spectrum of a bounded linear operator generalizes the notion of eigen values of the corresponding matrix, therefore, the study of spectrum of an operator takes a prominent position in solving many scientific and engineering problems. Hence, mathematicians and researchers have devoted their works in achieving new ideas and concepts in the concerned field. For instance, the fine spectrum of the Cesàro operator on the sequence space for has been studied by Gonzalez [1]. Okutoyi [2] computed the spectrum of the Cesàro operator over the sequence space . The fine spectra of the Cesàro operator over the sequence space have been determined by Akhmedov and Ba？ar [3]. Akhmedov and Ba？ar [4, 5] have studied the fine spectrum of the difference operator over the sequence spaces and , where . Altay and Ba？ar [6] have determined the fine spectrum of the difference operator over the sequence spaces , for . The fine spectrum of the difference operator over the sequence spaces and was investigated by Kayaduman and Furkan [7]. Srivastava and Kumar [8] have examined the fine spectrum of the generalized difference operator over the sequence space . Recently, the spectrum of the generalized difference operator over the sequence spaces and has been studied by Dutta and Baliarsingh [9, 10], respectively. The main focus of this paper is to define the difference operator and establish its spectral characterization with respect to the Goldberg’s classifications. Let be either constant or strictly