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

Each result cited is universally quantified over the data in its own statement.

Proof of claim 1. Let $R\in\mathbb{R}$ be positive and let $(\omega_{1},\omega_{2})$ be a structure pair for $F$ at $R$. The requirements that The First-Order Structure Condition for a Second-Order Equation Operator on a Hilbert Triple (i) and The Second-Order Structure Condition for an Equation Operator on a Hilbert Triple (ii) place on $\omega_{1}$ and on $\omega_{2}$ are the same, namely that $\omega_{1}$ be a modulus of continuity and that $\omega_{2}$ be a real-valued function on the pairs $(t,\alpha)$ with $0\le t$ and $1<\alpha$ for which $t\mapsto\omega_{2}(t,\alpha)$ is a modulus of continuity for each real $\alpha>1$; and the displayed inequality is the same in the two clauses. The two clauses differ only in the range of the form arguments: the first demands the inequality for all $X,Y\in\mathrm{Sym}(H)$, the second only for the pairs $(X,Y)$ of members of $\mathrm{Sym}(H)$ that are admitted at $\alpha$. Since every pair admitted at $\alpha$ is in particular a pair of members of $\mathrm{Sym}(H)$, each instance of the inequality demanded by the second clause is an instance of one supplied by the first, for the same $x,y\in W$, the same $r\in\mathbb{R}$ with $-R\le r\le R$ and the same $\alpha,\delta\in\mathbb{R}$ with $1<\alpha$ and $0<\delta<1$. Hence $(\omega_{1},\omega_{2})$ is a second-order structure pair for $F$ at $R$.

Suppose now that $F$ satisfies the first-order structure condition and let $R\in\mathbb{R}$ be positive. By The First-Order Structure Condition for a Second-Order Equation Operator on a Hilbert Triple (ii) there is a structure pair for $F$ at $R$, and by the previous paragraph it is a second-order structure pair for $F$ at $R$. As $R$ was an arbitrary positive real, $F$ satisfies the second-order structure condition by The Second-Order Structure Condition for an Equation Operator on a Hilbert Triple (iii).

Proof of claim 2. Let $(N_{m})_{m\in\mathbb{N}}$ be the tail forms of $(e_{k})_{k\in\mathbb{N}}$, which are defined because $(e_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $H$ all of whose members lie in $V$. Let $\beta,R,\rho\in\mathbb{R}$ be positive and let $\delta\in\mathbb{R}$ satisfy $0<\delta<1$; in particular $\delta$ is positive, so A First-Order Equation Operator is Degenerate Elliptic and Its $\delta$-Shifts Ignore the Form Argument (ii) applies to $F$ with this $\delta$.

For the condition of The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple (iii), let $(x_{m})_{m\in\mathbb{N}}$, $(r_{m})_{m\in\mathbb{N}}$, $(p_{m})_{m\in\mathbb{N}}$ and $(X_{m})_{m\in\mathbb{N}}$ be sequences as there. For each $m\in\mathbb{N}$ the forms $X_{m}$ and $X_{m}+\beta N_{m}$ both belong to $\mathrm{Sym}(H)$, by Hilbert Triples: Standing Notation and Background (vi), and $x_{m}\in W$, $r_{m}\in\mathbb{R}$ and $p_{m}\in H$; so A First-Order Equation Operator is Degenerate Elliptic and Its $\delta$-Shifts Ignore the Form Argument (ii), applied with these data and the two form arguments $X_{m}$ and $X_{m}+\beta N_{m}$, gives

$$F^{-}_{\delta}\bigl(x_{m},r_{m},p_{m},X_{m}\bigr)=F^{-}_{\delta}\bigl(x_{m},r_{m},p_{m},X_{m}+\beta N_{m}\bigr).$$

Since $0<\rho$, the right-hand side of the inequality required in The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple (iii) exceeds its left-hand side for every $m\in\mathbb{N}$, so that requirement holds with $m_{0}=1$.

For the condition of The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple (iv), let the four sequences be as there. The same claim of A First-Order Equation Operator is Degenerate Elliptic and Its $\delta$-Shifts Ignore the Form Argument, applied with the form arguments $X_{m}-\beta N_{m}$ and $X_{m}$, gives

$$F^{+}_{\delta}\bigl(x_{m},r_{m},p_{m},X_{m}-\beta N_{m}\bigr)=F^{+}_{\delta}\bigl(x_{m},r_{m},p_{m},X_{m}\bigr),$$

and again $0<\rho$ makes the required inequality hold for every $m\in\mathbb{N}$, so that requirement holds with $m_{0}=1$.

As $\beta$, $R$, $\rho$ and $\delta$ were arbitrary subject to the stated restrictions, $F$ is tail-insensitive along $(e_{k})_{k\in\mathbb{N}}$ by The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple (ii).

Proof of claim 3. Since $H$ is not finite-dimensional, An Orthonormal Basis of the Ambient Space Contained in the Form Space of a Hilbert Triple (i) provides an orthonormal basis $(e_{k})_{k\in\mathbb{N}}$ of $H$ with $e_{k}\in V$ for every $k\in\mathbb{N}$. By claim 2, $F$ is tail-insensitive along $(e_{k})_{k\in\mathbb{N}}$. Hence $F$ satisfies the tail-insensitivity condition by The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple (v).