A question on functional analysis

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 $$\langle(1 + T_2^{-1})^{-1} (\phi(x)), \phi (x)\rangle > \langle(1 + T_1^{-1})^{-1} (x), x\rangle, \forall x \in H_1$$

• Can you prove it in the special case when $H_1 = H_2$ and $\phi$ is the identity? If so, then try replacing $T_2$ with $\phi^{-1} T_2 \phi$. – Nate Eldredge Jun 1 '14 at 1:41
• You're assuming $T_{j}$ are invertible. Are you assuming $T_1 > 0$? – DisintegratingByParts Jun 1 '14 at 1:53
• @T.A.E. If you mean whether $T_1$ is a positive operator, I was not assuming that. But if assuming that helps, please tell me how, I will gain some insight into the problem.. – Amino Jun 1 '14 at 2:37
• @NateEldredge Unfortunately I cannot.... – Amino Jun 1 '14 at 2:37
• $(1+T^{-1})^{-1}=T(1+T)^{-1}$ is not well-defined if $-1\in\sigma(T)$. Is that another assumption for $T=T_{j}$? – DisintegratingByParts Jun 1 '14 at 3:37