= Square-root resolvent integral in a C-star algebra
{title2=$x^{1/2}=\pi^{-1}\int_0^\infty t^{-1/2}x(t1+x)^{-1}\,dt$}
For a <Positive element of a C-star algebra> $x$, the displayed improper integral converges in norm. Near zero its integrand has norm at most $t^{-1/2}$; near infinity it has norm at most $\|x\|t^{-3/2}$. The scalar identity follows by setting $t=su^2$ for $s>0$ and using $\int_0^\infty2/(1+u^2)\,du=\pi$; for $s=0$ both sides vanish. The bounds are uniform on the compact <spectrum of an element>, so <continuous functional calculus> gives the operator identity.
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