Is it possible to consider an approximation to a (non-self adjoint) operator with a self adjoint one? In operator theory it's wonderful if we have a self-adjoint operator (non necessarily bounded) due to all the work that has been done using their symmetry,... etc. I.e there are many powerful tools. 
My question is this: Can we consider any operator $T$ in the form 
$$ T = selfadjoint + nonselfadjoint $$
and maybe have some difference operator $D=T - T^*$ and say anything useful about it? Can we bound $D$ if we're in the appropriate space? Are there any papers anyone can recommend ?
 A: ? As you've suggested already yourself, is the answer not simply this: $T = S + D$ , with
$$
S = \frac{1}{2} \left( T + T^* \right) \qquad ; \qquad D = \frac{1}{2} \left(T - T^* \right)
$$
Where $S$ is self-adjoint and $D$ is a so-called anti self-adjoint operator: $D^* = -D$ .
In three dimensional space, the anti self-adjoint operator ($3 \times 3$ matrix)
has an interpretation as a momentary axis of rotation matrix : see my answer at
  Instantaneous Axis Of Rotation .
It can be easily shown that your $D$ must have the same form as the more common $\Omega$ in this answer, provided it's in three dimensional space:
$$
D = \Omega = \left[ \begin{array}{ccc}
0 & - \omega_z & +\omega_y \\ +\omega_z & 0 & -\omega_x \\ -\omega_y & +\omega_x & 0
\end{array} \right]
$$
The above splitting of $T$ in $S$ and $D$ is pretty standard in Fluid Dynamics, where
it gives rise to the symmetric strain rate tensor $S$
and the anti symmetric tensor $D$ (or rather $\Omega$ ) describing vorticity . Disclaimer: I don't really know how all this may be generalized to higher dimensions.
