Convergence of Second-Order Equation Operators on a Hilbert Triple on Bounded Test Data

In the setting of Hilbert Triples: Standing Notation and Background, the set $H$ is open in $H$ by Hilbert Triples: Standing Notation and Background §open-sets, with $W=D(A)$ there. Let $F$ and, for each $N\in\mathbb{N}$, $F_{N}$ be second-order equation operators on $H$ relative to $(H,V,A)$, with $\delta$-shifts $F^{-}_{\delta},F^{+}_{\delta}$ and $F^{-}_{N,\delta},F^{+}_{N,\delta}$ respectively, and let $\lVert\cdot\rVert$ be the norm on $\mathrm{Sym}(H)$ fixed in Hilbert Triples: Standing Notation and Background §restriction.

1. (Bounded test data)¶ For a real number $R>0$, a test datum bounded by $R$ is a quadruple $(x,r,p,Y)$ with $x\in D(A)$, $r\in\mathbb{R}$, $p\in H$ and $Y\in\mathrm{Sym}(H)$ such that

$$|x|_{V}\le R,\qquad|r|\le R,\qquad|p|_{H}\le R,\qquad\lVert Y\rVert\le R .$$

2. (Convergence)¶ The sequence $(F_{N})_{N\in\mathbb{N}}$ converges to $F$ on bounded test data if for all real numbers $\delta>0$, $R>0$ and $\varepsilon>0$ there is $N_{0}\in\mathbb{N}$ such that

$$\bigl|F^{-}_{N,\delta}(x,r,p,Y)-F^{-}_{\delta}(x,r,p,Y)\bigr|\le\varepsilon\quad\text{and}\quad\bigl|F^{+}_{N,\delta}(x,r,p,Y)-F^{+}_{\delta}(x,r,p,Y)\bigr|\le\varepsilon$$

for every $N\in\mathbb{N}$ with $N\ge N_{0}$ and every test datum $(x,r,p,Y)$ bounded by $R$.