Let $H$ be a Hilbert space. Suppose that $T:H \rightarrow H$ is a $0.5$-averaged nonexpansive mapping and $T_N:H \rightarrow H$ is nonexpansive too. T is given as $T=0.5I+0.5T_N$. Prove that the mapping $N = 2T - I$ is nonexpansive, and relate the set offixed points of $T$ with those of $N$.
$N$ is nonexpansive: $N = 2T - I = 2((1-0.5)I + 0.5T_N) - I = T_N$ is nonexpansive clearly, since $T_N$ is nonexpansive by definition of T.
But how to relate the fixed point sets? $\big\{ T(x)=x \big\} \iff \big\{ N(x)=2x-I \big\}$ ?
Thanks in advance!