Essential Numerical Range and Davis-Wielandt Shell

The study of the Davis-Wielandt shell forms a very important generalization of the numerical range in functional analysis. Hiroshi Nakazito and Mao-Ting Chien studied the connections between the q-numerical range and the Davis-Wielandt shell. Chi-Kwong Li and Yiu-Tung Poon studied the boundary of the Davis-Wielandt shells of normal operators. However the characterization of the essential numerical range We(T) and the Davis-Wielandt shell DW(T) has not been exhausted. One of the pending questions that remained was: What are the connections between the We(T) and the DW(T) of an operator? Moreover what are the conditions when We(T) and the classical numerical range W(T) coincide in the Davis-Wielandt shell? Therefore we have presented the Davis-Wielandt shells and the essential numerical range of operators in Hilbert spaces. In this study we have investigated the following; the relationship between the DW(T) of an operator and the We(T); the relationship between the essential spectrum and the DW(T) of an operator; the condition when the We(T) and the W(T) coincide in the Davis-Wielandt shell.