I don't get it why the parts I circled in red hold
I know that both $A_k$ and $\mathcal{I} - A_k$ are positive but I don't think that information let us conclude directly what's in the red boxes.
can someone please give more details why those inequalities hold ?
by the way a positive operator is an operator whose associated quadratic form is positive definite and in addition to that is self adjoint.
