The Resolvent of the Form Operator of a Hilbert Triple: Minimisation, Contraction, and Density of the Domain

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, and let $(H,V,A)$ be a Hilbert triple, with the inner products, norms and distances of $H$ and $V$ written with the subscripts $H$ and $V$, and with $D(A)$ and $A$ as in Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iv). Write $\tfrac12$ for $2^{-1}$. Let $\mathbb{N}$ be the set of natural numbers, read in $\mathbb{R}$ through the canonical map as in The Real Numbers and Standard Notation, so that $0<k$ in $\mathbb{R}$ for every $k\in\mathbb{N}$ by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; sequences are indexed by $\mathbb{N}$. Let $\alpha\in\mathbb{R}$ with $0<\alpha$; since $\alpha$ is arbitrary, each claim below holds for every positive real number in place of $\alpha$, and for a positive real $\beta$ we write $R_{\beta}$ for the map of claim 1 with $\beta$ in place of $\alpha$. For $\bar{x}\in H$ define $\Phi_{\bar{x}}:V\to\mathbb{R}$ by

$$\Phi_{\bar{x}}(x)=\tfrac12|x|_{V}^{2}+\alpha\,|x-\bar{x}|_{H}^{2}\qquad(x\in V).$$

Then the following hold.

1. (Minimiser)(i) For every $\bar{x}\in H$ there is exactly one point of $V$, denoted $R_{\alpha}\bar{x}$, such that $\Phi_{\bar{x}}(R_{\alpha}\bar{x})\le\Phi_{\bar{x}}(x)$ for every $x\in V$.

2. (Euler-Lagrange equation)(ii) For every $\bar{x}\in H$, $R_{\alpha}\bar{x}\in D(A)$ and $A(R_{\alpha}\bar{x})=2\alpha(\bar{x}-R_{\alpha}\bar{x})$. Conversely, if $x\in D(A)$ satisfies $Ax+2\alpha x=2\alpha\bar{x}$, then $x=R_{\alpha}\bar{x}$.

3. (Bounds)(iii) For every $\bar{x}\in H$, $\tfrac12|R_{\alpha}\bar{x}|_{V}^{2}+\alpha|R_{\alpha}\bar{x}-\bar{x}|_{H}^{2}\le\alpha|\bar{x}|_{H}^{2}$; and if $\bar{x}\in V$, then $\tfrac12|R_{\alpha}\bar{x}|_{V}^{2}+\alpha|R_{\alpha}\bar{x}-\bar{x}|_{H}^{2}\le\tfrac12|\bar{x}|_{V}^{2}$, so that $|R_{\alpha}\bar{x}|_{V}\le|\bar{x}|_{V}$ and $|R_{\alpha}\bar{x}-\bar{x}|_{H}^{2}\le|\bar{x}|_{V}^{2}/(2\alpha)$.

4. (Contraction)(iv) For all $\bar{x},\bar{y}\in H$, $|R_{\alpha}\bar{x}-R_{\alpha}\bar{y}|_{H}\le|\bar{x}-\bar{y}|_{H}$.

5. (Approximation)(v) For every $\bar{x}\in H$ and every real $\varepsilon>0$ there is a real $\alpha_{0}>0$ such that $|R_{\beta}\bar{x}-\bar{x}|_{H}<\varepsilon$ for every real $\beta\ge\alpha_{0}$.

6. (Density in $H$)(vi) $D(A)$ is dense in $H$.

7. (Density in $V$)(vii) Suppose that $(V,d_{V})$ is separable. Then for every $\bar{x}\in V$ the sequence $(R_{k}\bar{x})_{k\in\mathbb{N}}$, where $R_{k}$ is the map of claim 1 with the positive real number $k$ in place of $\alpha$, converges to $\bar{x}$ in $(V,d_{V})$; consequently $D(A)$ is dense in $V$ for the metric $d_{V}$.