= Spectral positivity criterion for a self-adjoint operator
{title2=$A\geq0\Longleftrightarrow\sigma(A)\subseteq[0,\infty)$}
<Positivity> makes $A-z$ bounded below with dense closed range for every $z<0$, excluding negative spectral values. Conversely if the <spectrum> is nonnegative, $R=(I+A)^{-1}$ has <spectrum> in $[0,1]$ by the <resolvent spectral mapping identity>. The <numerical-range endpoint approximate-eigenvalue theorem> gives $0\leq R\leq I$, and positive-form <Cauchy-Schwarz inequality> gives $\|Rf\|^2\leq\langle Rf,f\rangle$. For $u=Rf$, $\langle Au,u\rangle=\langle Rf,f\rangle-\|Rf\|^2\geq0$.
Back to article page
