First-Order Operators Satisfy the Second-Order Structure and Tail-Insensitivity Conditions

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 $\mathrm{Sym}(H)$ be as in Hilbert Triples: Standing Notation and Background §restriction, and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$. Then the following hold.

1. (A first-order structure pair serves at second order)¶ Let $R\in\mathbb{R}$ be positive and let $(\omega_{1},\omega_{2})$ be a structure pair for $F$ at $R$. Then $(\omega_{1},\omega_{2})$ is a second-order structure pair for $F$ at $R$. Consequently, if $F$ satisfies the first-order structure condition, then $F$ satisfies the second-order structure condition.

2. (First-order operators are tail-insensitive)¶ Assume that $F$ is first order, and let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$ with $e_{k}\in V$ for every $k\in\mathbb{N}$. Then $F$ is tail-insensitive along $(e_{k})_{k\in\mathbb{N}}$.

3. (The tail-insensitivity condition for a first-order operator)¶ Assume that $F$ is first order and that $H$, as a vector space over $\mathbb{R}$, is not finite-dimensional. Then $F$ satisfies the tail-insensitivity condition.