I’m looking for examples of automorphisms. I came up with a curious one, but I don’t know if it is correct:
Let $\mathbb{C}^{\infty}$ be the space of smooth complex functions $\tau:\Omega\rightarrow\mathbb{C}$ defined over the domain $\Omega$, and $f\in\mathbb{C}^{\infty}$. Define the linear endomorphism $T:\mathbb{C}^{\infty}\rightarrow\mathbb{C}^{\infty}$ as: $$T(f)=\frac{\mathrm{d}f}{\mathrm{d}z}=g(t)\quad \{z;t\}\in \Omega$$ Define $T^{-1}:\mathbb{C}^{\infty}\rightarrow\mathbb{C}^{\infty}$ to be: $$T^{-1}(g)=\int_{t_1}^{t_2}g(t)\mathrm{d}t=f(z)$$ Since: $$T\circ T^{-1}(f)=f$$ Then, $T$ is an automorphism.