Let $H$ be a Hilbert space, and let $T \colon H \to H$ be a bijective bounded linear operator whose inverse is bounded. Then how to show that $(T^*)^{-1}$ exists and $$(T^*)^{-1} = (T^{-1})^*?$$
My effort:
Let $x, y \in H$ such that $y = Tx$ and $T^* y = 0$. Then we have $$0 = \langle T^* y , x \rangle = \langle y, Tx \rangle = \langle y, y \rangle,$$ which implies that $y = 0$. Thus $T^*$ is injective.
Can we now say that $(T^*)^{-1}$ exists?
Is it possible now to show that $T^* \colon H \to H$ is surjective?
Finally, how to derive the relation $$(T^*)^{-1} = (T^{-1})^*?$$