On Joint Essential Numerical Ranges

Abstract

The concept of numerical range on a Hilbert space was first introduced by O. Toeplitz in 1918 for matrices. This notion was independently ex tended by G. Lumer and F. Bauer in the sixties on finite dimensional Banach spaces. J. G. Stampfli introduced the maximal numerical range, proved its convexity and used it to derive an identity for the norm of derivation in 1970. In 1972, J. G. Stampfli and J. P. Williams defined and studied the essential numerical range of an operator. In our work, we looked at the joint essential numerical ranges. In particular, this study has shown that the properties of numerical ranges such as com pactness, nonemptiness and convexity do hold for the joint essential nu merical range. The study has also shown that the closure of the joint numerical range of an operator is star-shaped with elements in the joint essential numerical range of the operator as star centers. Further, we have shown that the joint essential spectrum is contained in the joint essen tial numerical range by looking at the boundary of the joint spectrum. Convexity, nonemptiness and compactness of the joint essential numerical range were shown by first proving the equivalent definitions of the joint essential numerical range. Basing on the convexity of the joint essential numerical range, other results were obtained. The results of this study are helpful in the development of the research on numerical ranges and may also be applied by mathematicians in solving several problems in operator theory.