On Essential Numerical Range and Davis-Wielandt Shell of Hilbert Space Operators

Let the numerical range of the operator T be denoted by W (T), the essential numerical range by We (T) and the Davis-Wielandt shell by DW (T). We discuss the equality of the essential numerical range and the numerical range of the operator T in a complex infinite dimensional Hilbert space H. The results obtained are used to study the instance when the essential numerical range can replace the numerical range in the Davis-Wielandt shell.