I came across this exercise from book 'Linear Operator Theory in Engineering and Science'. I want to show that function multiplication is unitary if and only if the absolute value of the function is 1 a.e.. The problem is below: $$ \begin{array}{l}{\text { Let I }=[a, b] \text { be a bounded interval and define } F: L_{2}(I) \rightarrow L_{2}(I)} \\ {\text { by }(F x)(t)=f(t) x(t), \text { where } f \in L_{\infty}(I) . \text {how toshow that } F \text { is a unitary mapping if and only if }|f(t)|=1 \text { almost }} \\ {\text { everywhere? }}\end{array} $$
Any help is appreciated.