Here is the situation:
I have the operator/matrix expression AB+BA and want to know the norm.
1.) B is compact, so I believe AB+BA is compact (as are AB and BA, so any finite-dimensional matrix answer should be fine, or at least informative)
2.) A is both unitary and self-adjoint (and therefore A^2 = I)
3.) AB+BA itself is self-adjoint, because I also know that AB and BA are self-adjoint.
4.) I know the norm, eigenvalues, basically everything about both A and B (and A* and B*), AND I know all of this about AB and BA (and their adjoints) as well. An expression in terms of the norms/eigenvalues of those four operators would give me everything I want.
5.) A sharp inequality would be okay but really I need to find the exact expression for the norm or the max eigenvalue.