Locally Strictly Proper Second-Order Equation Operator on a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be open in $H$, with $W=D(A)\cap U$ as in Hilbert Triples: Standing Notation and Background §open-sets, let $\mathrm{Sym}(V)$ be as in Hilbert Triples: Standing Notation and Background §restriction, and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$.

1. (Properness constant at a level)¶ Let $R,\lambda\in\mathbb{R}$ be positive. We say that $\lambda$ is a properness constant for $F$ at $R$ if

for every $x\in W$, every $p\in H$, every $X\in\mathrm{Sym}(V)$ and all $r,s\in\mathbb{R}$ with $-R\le s\le r\le R$.

2. (Local strict properness)¶ The operator $F$ is locally strictly proper if for every positive $R\in\mathbb{R}$ there is a properness constant for $F$ at $R$.