On the similarities in properties of essential numerical range and Davis-Wielandt shell of Hilbert space operators

Abstract

Let be an operator on an infinite dimensional Hilbert space . Denote the essential numerical range of the operator by and the Davis-Wielandt shell of the operator by . We review the properties of the essential numerical range and those of the Davis-Wielandt shell. This review is aimed at striking similarities in the properties shared. The results of this study show that some of the properties shared are, for instance, unitary invariance and convexity. However, it is noted that the Davis-Wielandt shell is convex if . If , then is an ellipsoid. Thus is convex or encloses a convex set. It is also shown that both and behave in the same way if is the identity operator and scalar multiplication. This review is a contribution to operator theory and the area of numerical ranges.