On the Decomposition of Hilbert Spaces

A basic relation between numerical range and Davis-Wielandt shell of an operator A acting on a Hilbert space with orthonormal basis xi = {ei|i is an element of I} and its conjugate (A) over bar which is introduced in this paper are obtained. The results are used to study the relation between point spectrum, approximate spectrum and residual spectrum of A and A ((A) over bar A necessary and sufficient condition for A to be self-conjugate (A = (A) over bar) is given using a subgroup of H.