# spectral theorem : the case of bounded operator

Let $H$ be Hilbert space, and $T$ be a self-adjoint operator. Then, by spectral theorem, there exists $\{E_{\lambda}\}$ s.t., $Tx=\int _{-\infty}^{\infty}\lambda dE_{\lambda}x$

Question: If $||T||\leq 1$, we can represent $Tx=\int _{-1}^{1}\lambda dE_{\lambda}x$? I tried to calculate $||Tx||^2$ but I couldn't prove it.

Here $\left\{E_{\lambda}\right\}$ is the projection valued measure associated to the operator $T$ by the spectral theorem. It is known that in this case the support of $\left\{E_{\lambda}\right\}$ (as a p.v.m.) is given by $$\operatorname{supp}\left\{E_{\lambda}\right\}=\sigma(T)$$ Since $\|T\|\leq 1$ and $T$ is self-adjoint, then $\sigma(T)\subset [-1,1]$. Therefore $\operatorname{supp}\left\{E_{\lambda}\right\}\subset [-1,1]$ and so $$T=\int_{-\infty}^{+\infty}\lambda dE_{\lambda}=\int_{-1}^{1}\lambda dE_{\lambda}$$