Computing eigenspectrum of $T:L^2[0,1]\to L^2[0,1]$, defined by $(Tf)(t)=tf(t)$

Question

I have to compute the eigenspectrum of operator $T:L^2[0,1]\to L^2[0,1]$, defined by $(Tf)(t)=tf(t)$. If $\lambda$ is an eigenvalue, then there is a nonzero $f$ such that $Tf=\lambda f$ $\Rightarrow$ $tf(t)=\lambda f(t)$, $\forall t\in [0,1]$. But as $f$ is nonzero, it takes nonzero values on a set of some positive measure. This means $\lambda$ will be equal to more than one $t\in [0,1]$, which is a contradiction. So, this means eigenspectrum should be empty. Am I correct.