If A and B are positive definite matrices, is it true that the (induced, L2) matrix norm of A times inv(A+B)

Question

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?

Answer 1

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

Answer 2

No. Random counterexample: $$ A = \pmatrix{5&4\\ 4&5}, \ B = \pmatrix{2&3\\ 3&5}. \ C=A(A+B)^{-1}=\frac1{21}\pmatrix{22&-7\\ 5&7}. $$ The first column of $C$ is clearly longer than a unit vector. Hence $\|C\|>1$.