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?

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 $\|C\|>1$.
