Given that, on an arbitrary vector space $V$ over a field $K$ (where $K$ is either the real or the complex numbers) with inner product, the linear operator $T$ is such that $\langle T(v),w\rangle= \langle v,T(w)\rangle$, we must be able to prove that the linear operator on $V$ given by $(1_V-iT)$ is bijective.
Honestly, I have very little idea how to prove this. I would like to think that by proving injectivity, since the operator is linear then we also have surjectivity. I do not know, however, how the hypothesis for the product comes in.
Also, maybe if $T$ is bijective, then the other operator must also be bijective.