Characterization of Essential Numerical Rangers and Davis-Wielandt Shells of Hilbert Space Operators.

Abstract

The study of the Davis- Wielandt shell forms a very important generaralization of the numerical range in functional analysis. Hiroshi Nakazito and Mao-Ting Chien studied the connections between the q-numericol range and the Davis-Wielandt shell. Chi-Kwong Li and Yiu-Tung Poon studied the boundary of the Davis-Wiolandt shells of normal operators. How• ever, the characterisation of the essential numerical ranqe, Wc(T), and the Davis- Wielandt shell, DW (T) has not been exhausted. One of the pending questions that. remained was: What. are the connections bet.ween the W0(T) and the DW(T) of an opera.tor? Moreover, what are the condi• tions 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 t.he following; the relationship between the DW(T) of an operator and the W,(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- Wieland! shell. The method• ology involved the use of inner product spaces, the Cauchy-Schwarz and triangle inequalities. The results of this study showed that the essential numerical range and the Davis- wieiond: shells of an operator share a va- riety of properties, for instance, identity property and unitary invariance. It. was also noted that. the essential spectrum is contained in the closure of the first co-ordinate of the Davis- Wielandt shell. Moreover, the We(T) and the W(T) coincide in DW(T) if and only if T = >J. The results obtained would be useful in applications involving systems of differential equations and aerodynamics.