Verify if we can find a function $\beta(x) \in C^\infty$ such as the operators $T_{\pm}$ defined as
\begin{equation} (T_{+}f)(x) = \beta(x)[ \beta(x)f(x)+\beta(-x)f(-x)] \end{equation} \begin{equation} (T_{-}f)(x) = \beta(-x) \left[ \beta(-x)f(x)-\beta(x)f(-x)\right] \end{equation}
are orthogonal projectors $(P^2=P=P^*)$ and satisfies the follow
\begin{equation} \beta(x) = \left\lbrace \begin{array}{ll} x>1 & 1\\ x<-1 & 0 \end{array} \right. \end{equation} \begin{equation} T_+T_-=T_-T_+=0 \end{equation} \begin{equation} T_++T_-=I \end{equation}
I've been trying to obtain something using the definition of orthogonal projectors and even trying to represent everything as matrix but i don't really get anything, i think it's a problem of misconception but i've been stuck with this problem for days reading and reading about this subject, help please!