I am struggling with the following question.
Let $V$ be a finite dimensional complex inner product space and let $f:V\to V$ be a self-adjoint linear map. Show that the map $f-i\cdot \text{id}_v$ is invertible and that the linear map $g$ given by $$g=(f+i\cdot \text{id}_v)(f-i\cdot \text{id}_v)^{-1}$$ is unitary by verifying that $g^*g=\text{id}_v$.
I assume that using the inverse of $f-i\cdot \text{id}_v$ is useful in verifying $g^*g=\text{id}_v$. I also imagine that for the second part using the fact that $(fg)^*=g^*f^*$ and $(f^*)^{-1}=(f^{-1})^*$ for linear maps $f$ and $g$ might be useful.