The Banach Numerical Range for Finite Linear Operators

Abstract

The numerical range has been a subject of interest to many researchers and scholars in the recent past. Based on the research outputs, many results have been obtained. Besides, several generalizations of the classical numerical range have also been made. The recent developments have focused on the theory of operators on Hilbert spaces. The determination of the numerical ranges of linear and nonlinear operators have been given in both the Hilbert and Banach spaces. In addition, results of these numerical ranges have been extended to the case of two operators in both spaces. It is important to note that more generalizations have been made in Hilbert spaces as compared to those that have been made in the Banach spaces. The Banach space has two major numerical ranges which are: the spatial and algebraic numerical ranges. This research focuses on determining the numerical range for a finite number of linear operators in the Banach space based on the classical definition. Properties which hold for the classical numerical range have been shown to hold for the Banach space numerical range. The property of convexity has been established using the Toeplitz-Hausdorff theorem under the condition that the Banach space is smooth. Furthermore, the numerical radius and the spectrum of these operators have also been determined.