Find out if there is an unitary operator

Question

I need to find out if there is an unitary operator $T$:$\mathbb C^2$$\rightarrow$$\mathbb C^2$ that satisfies $T(1,1)=e^{i(2+i)}(1,1)$

I don't really know how to approach this problem so any help is appreciated.

Answer 1

Unitary operators preserve the norm. Here $\|T(1,1)\|=|e^{i(2+i)}|\|(1,1)\|=e^{-2}\|(1,1)\|$ so $\|T(1,1)\| \neq \|(1,1)\|$.

Answer 2

If $ \lambda$ is an eigenvalue of an unitary operator, then $ |\lambda|=1.$

Your operator $T$ has the eigenvalue $e^{i(2i+1}$, but $|e^{i(2i+1}|=e^{-2}$

Conclusion ?