If A and B are positive definite matrices, is it true that the (induced, L2) matrix norm of A times inv(A+B) If $A$ and $B$ are positive definite matrices, can it be shown that $||A (A + B)^{-1}|| \leq 1$, where $||...||$ is the matrix norm induced by the $L_2$ norm on vectors?

It is true, if $A$ and $B$ commute, as: 
$$ A(A+B)^{-1}(A+B)^{-1} A = (I+\underbrace{A^{-1}BA^{-1}}_{\text{pos. def.}})^{-2}. $$
