Classification of compact multiplication operators

Question

It is known, that if $A_\phi:L_2[0,1] \to L_2[0,1]$ - multiplication operator defined as $(A_\phi x)(t)=\phi(t) \cdot x(t)$ is compact $\Rightarrow$ $\phi = 0$ $a.e.$ I wonder if this is also true for $A_\phi:L_2[0,1] \to L_1[0,1]$, where $A_\phi$ is defined the same way.