Let $H$ be a Hilbert space over $\mathbb{C}$ and $T$ be a self-adjoint operator. Suppose there exists $c>0$ s.t., for every $x\in H$, $(Tx,x)\geq c(x,x)$ Then $S:=\int_{-\infty}^\infty (y^2+T^2)^{-1} \, dy$ is well-defined, and represent $S$ specifically.
I know that spectrum theorem $T=\int \lambda \, dE_\lambda$. But I can't prove well-defined of $S$ and represent.