Is this operator continuous?

Question

Let $$ f(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } x \leq 0 \\ -3 & \mbox{if } 0<x \leq 1 \\ 1/x & \mbox{if } x > 1 \\ \end{array} \right. \\ $$ and let the operator $T(g) = fg$ in the space $L_2(\mathbb{R})$.

I want to know if the operator is continuos and how I can get $\|T\|$.

Answer 1

Since $||f||_{\infty} = 3$, we get $||T(g)|| \leq 3||g||$. For $g = I_{[0,1]}$, the indicator function of $[0,1]$, we have equality. Hence $||T|| = 3$.