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 $\xi=\{e_{i}|i \in I\}$ and its conjugate $\bar{A}$ 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 $\bar{A}$. A necessary and sufficient condition for $A$ to be self-conjugate ($A=\bar{A}$) is given using a subgroup of H.