A question on functional analysis

Question

Let $H_i$, where $i = 1,2$ be Hilbert spaces and $T_i : H_i \rightarrow H_i$ be closed operators, such that $T_i$ have positive spectrum. Let $\phi : H_1 \rightarrow H_2$ is an isometric isomorphism and suppose $$\left\langle T_2 (\phi (x)) , \phi (x)\right\rangle > \left\langle T_1 (x) , x\right\rangle, \forall x \in H_1$$ We need to prove that $$\langle(1 + T_2^{-1})^{-1} (\phi(x)), \phi (x)\rangle > \langle(1 + T_1^{-1})^{-1} (x), x\rangle, \forall x \in H_1$$