Define \begin{eqnarray*} G_{\lambda}(|\xi|^{2}) = \left\{ \begin{array}{lcl} 0, & & \lambda < 0 \\ ~ \\ \chi_{_{\lbrace |\xi|^{2} \leq \lambda \rbrace }}, & & \lambda \geq 0, \end{array} \right. \end{eqnarray*} and $$ E_{0}(\lambda)f = \mathcal{F}^{-1}[G_{\lambda}(|\xi|^{2})\mathcal{F}(f)], \ \ f \in L^{2}(\mathbb{R}^{n}) $$ How do I show that

$$(E_{0}(\lambda)f|f) = \|E_{0}(\lambda)f\|^{2}_{L^{2}}, \ \ f \in L^{2}(\mathbb{R}^{n})$$

Let $\lambda>0$ be given. Define $$G_{\lambda}(\xi) = \chi_{_{\lbrace |\xi|^{2} \leq \lambda \rbrace }}. $$ and $$ E_{0}(\lambda)f = \mathcal{F}^{-1}[G_{\lambda}(|\xi|^{2})\mathcal{F}(f)], \ \ f \in L^{2}(\mathbb{R}^{n}) $$ How do I show that

$$(E_{0}(\lambda)f|f) = \|E_{0}(\lambda)f\|^{2}_{L^{2}}, \ \ f \in L^{2}(\mathbb{R}^{n})\qquad ?$$