necessary and sufficient conditions for multiplication operator T on L^([a,b]) to be a projection

Question

Let $T$ be a multiplication operator on $L^2([a, b ])$. Find necessary and sufficient conditions for $T$ to be a projection. let g be a fixed function in $L^2([a,b])$, and $T(f(x))=g(x)f(x)$.

Answer 1

T is a projection, iff $T^2 = T$. So $g^2(x)f(x) = g(x)f(x)$ for each $f\in L^2([a,b])$. Thus it must be $g^2(x)-g(x)=0$ (which means $g(x)\in\{0,1\}$) almost everywhere.