On $L^2(-\infty, \infty)$, T is a bounded linear operator and is defined by $$Tf(t)= \begin{cases} f(t), & \text{for } t \geq 0 \ \cr -f(t), & \text{for } t<0, \end{cases}$$
I'm able to prove $T$ is unitary. But then how to determine the spectrum and spectral representation of T?
Since $T$ is unitary, its spectrum lies in the unit circle. Since $T$ is selfadjoint, the spectrum lies in the real line. So the only possible elements of the spectrum are $-1,1$ (you can also deduce this from the fact that $T^2=I$).
Both values in the spectrum are eigenvalues, as you can see with the characteristic functions of $[0,\infty)$ and $(-\infty,0)$.
The spectral decomposition of $T$ is $T=P-(I-P)$, where $P$ is the projection $$ Pf(t)=\begin{cases}f(t),&\ t\geq0\\ 0,&\ t<0\end{cases} $$