Prove or disprove that $g\in L(V,V)$ exists with $f=g\circ g^{ad}$.

Question

Let $V=\mathbb C^{2,1}$ with standard scalarproduct and $f\in L(V,V)$ with $f\biggr( \begin{bmatrix} x \\ y \end{bmatrix} \biggl)=\begin{bmatrix} x +2y \\ 2x+y \end{bmatrix}$ Prove or disprove that $g\in L(V,V)$ exists with $f=g\circ g^{ad}$.

I do not see how to handle this problem. I would start to try to find such $g$ like this, set $\begin{bmatrix} x +2y \\ 2x+y \end{bmatrix}=F$, $v=(x,y)^T, w=(a,b)^T$ then, $$ \langle f(v),w\rangle=w^HF=\langle g^{ad}(v),g^{ad}(w)\rangle$$, but how should one continue. Could someone help me here and tell me how to attack this problem?

P.S.: If needed, in the definition I am using I have linearity in the second component and semi-linearity in the first.

Answer 1

Hint:

The matrix of $f$ with respect to the standard basis is $\pmatrix{1 & 2 \\ 2 & 1}$.

This matrix is not positive because $$\det\pmatrix{1 & 2 \\ 2 & 1} = -3 < 0$$ However, the map $g\circ g^{ad}$ is positive for any linear map $g$.