Let T be a bounded operator on a Hilbert Space H. I am trying to show that $T + T^* \geq 0$ iff $T + I$ is invertible and $\left\vert\left\vert{(T - I)(T + I)^{-1}}\right\vert\right\vert \leq 1$.
I have a hint which says prove $T + T^* \geq 0$ iff $\left\vert\left\vert{(T + I)x}\right\vert\right\vert \geq \left\vert\left\vert{x}\right\vert\right\vert$ and $\left\vert\left\vert{(T + I)x}\right\vert\right\vert \geq \left\vert\left\vert{(T - I)x}\right\vert\right\vert$ for every $x \in H$.
I am able to prove the hint. I am also able to show that $T + T^* \geq 0$ implies that $T + I$ is invertible. I am struggling trying to show that $\left\vert\left\vert{(T - I)(T + I)^{-1}}\right\vert\right\vert \leq 1$ and $\left\vert\left\vert{(T + I)x}\right\vert\right\vert \geq \left\vert\left\vert{x}\right\vert\right\vert$.
Any ideas or tips would be greatly appreciated.
If you know that $\|(T-I)x \|\leq \|(T+I)x \|$ for all $x \in H$, then you have that $$\|(T-I)(T+I)^{-1}x \| \leq \|(T+I)(T+I)^{-1}x \|= \|x \|, \ x \in H. $$ Hence $\|(T-I)(T+I)^{-1}\| \leq 1 $.
On the other hand, note that if $T+T^*\geq 0$ then $$ 2\mbox{Re}(\langle Tx,x \rangle)=\langle Tx,x \rangle+\langle x,Tx \rangle=\langle Tx,x \rangle+\langle T^*x,x \rangle =\langle Tx+T^*x,x \rangle \geq 0$$ for all $x \in H$. Then we have $$\|(T+I)x \|^2 = \|Tx \|^2+2\mbox{Re}(\langle Tx,x \rangle) + \|x \|^2\geq \|x \|^2, \ x \in H. $$ Hence $$\|(T+I)x \| \geq \|x \|, \ x \in H.$$