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

We use the identities of Elementary Identities in a Real Inner Product Space, The Cauchy–Schwarz Inequality in a Real Inner Product Space, The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity, the triple facts Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator, and for real numbers Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field, Properties of the Absolute Value in an Ordered Field and Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Condition (a) of the triple is $|x|_{H}\le|x|_{V}$ on $V$ and condition (b) is the density of $V$ in $H$. Throughout, $\alpha>0$ is fixed and $\bar{x}\in H$; the constructions are repeated verbatim for any positive real in place of $\alpha$.

An equivalent inner product on $V$. For $x,y\in V$ put $[x,y]=\langle x,y\rangle_{V}+2\alpha\langle x,y\rangle_{H}$. This is symmetric, additive and homogeneous in the first argument as a sum of two such maps, and $[x,x]=|x|_{V}^{2}+2\alpha|x|_{H}^{2}\ge|x|_{V}^{2}\ge 0$ with $[x,x]=0$ only if $x=0_{H}$; so $[\cdot,\cdot]$ is an inner product on $V$. Write $\lVert x\rVert_{\ast}$ for the associated norm; then $|x|_{V}\le\lVert x\rVert_{\ast}$ and, since $|x|_{H}\le|x|_{V}$ gives $|x|_{H}^{2}\le|x|_{V}^{2}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), hence $2\alpha|x|_{H}^{2}\le 2\alpha|x|_{V}^{2}$ (claim 5 of Elementary Arithmetic in an Ordered Field) and, adding $|x|_{V}^{2}$ to both sides (order axiom of Ordered Field), $\lVert x\rVert_{\ast}^{2}\le(1+2\alpha)|x|_{V}^{2}$. Consequently $(V,[\cdot,\cdot])$ is a real Hilbert space: a Cauchy sequence for $\lVert\cdot\rVert_{\ast}$ is Cauchy for $|\cdot|_{V}$, hence converges in $(V,d_{V})$ to some $x$, and $\lVert x_{m}-x\rVert_{\ast}^{2}\le(1+2\alpha)|x_{m}-x|_{V}^{2}$ shows that it converges to $x$ for $\lVert\cdot\rVert_{\ast}$ as well (choose $N$ with $|x_{m}-x|_{V}^{2}<\varepsilon^{2}/(1+2\alpha)$ for $m\ge N$).

The representing vector. The map $\ell(y)=2\alpha\langle\bar{x},y\rangle_{H}$ on $V$ is linear (Elementary Identities in a Real Inner Product Space (i)) and $|\ell(y)|\le 2\alpha|\bar{x}|_{H}|y|_{H}\le 2\alpha|\bar{x}|_{H}\lVert y\rVert_{\ast}$ by Cauchy-Schwarz, $|y|_{H}\le|y|_{V}\le\lVert y\rVert_{\ast}$ and claim 5 of Elementary Arithmetic in an Ordered Field; so $\ell$ is a bounded linear functional on $(V,[\cdot,\cdot])$. By The Riesz Representation Theorem for a Real Hilbert Space (i) and The Riesz Representation Theorem for a Real Hilbert Space (ii) there is a unique $\hat{x}\in V$ with $[y,\hat{x}]=\ell(y)$ for all $y\in V$, that is, by symmetry and bilinearity,

$$\langle\hat{x},y\rangle_{V}+2\alpha\langle\hat{x},y\rangle_{H}=2\alpha\langle\bar{x},y\rangle_{H},\qquad\text{equivalently}\qquad\langle\hat{x},y\rangle_{V}=\langle 2\alpha(\bar{x}-\hat{x}),y\rangle_{H}\qquad(y\in V).$$

We refer to this pair of identities as (E).

Claims 1 and 2. Let $x\in V$ and put $w=x-\hat{x}\in V$. By Elementary Identities in a Real Inner Product Space (v) applied in $V$ and in $H$,

$$\Phi_{\bar{x}}(x)-\Phi_{\bar{x}}(\hat{x})=\langle\hat{x},w\rangle_{V}+\tfrac12|w|_{V}^{2}+2\alpha\langle\hat{x}-\bar{x},w\rangle_{H}+\alpha|w|_{H}^{2}=\tfrac12|w|_{V}^{2}+\alpha|w|_{H}^{2},$$

the first two inner-product terms cancelling by (E) with $y=w$ (note $\langle\hat{x}-\bar{x},w\rangle_{H}=-\langle\bar{x}-\hat{x},w\rangle_{H}$). The right side is nonnegative and vanishes only if $|w|_{V}=0$, that is, $x=\hat{x}$. Hence $\hat{x}$ satisfies $\Phi_{\bar{x}}(\hat{x})\le\Phi_{\bar{x}}(x)$ for all $x\in V$, and any $x$ with this property has $\Phi_{\bar{x}}(x)-\Phi_{\bar{x}}(\hat{x})\le 0$, forcing $x=\hat{x}$: this proves claim 1 with $R_{\alpha}\bar{x}=\hat{x}$. By (E) and Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iv), $\hat{x}\in D(A)$ with $A\hat{x}=2\alpha(\bar{x}-\hat{x})$. Conversely, if $x\in D(A)$ and $Ax+2\alpha x=2\alpha\bar{x}$, then $\langle x,y\rangle_{V}=\langle Ax,y\rangle_{H}=\langle 2\alpha(\bar{x}-x),y\rangle_{H}$ for all $y\in V$, so $x$ satisfies the identity characterising $\hat{x}$ and $x=\hat{x}$ by the uniqueness of the representing vector. This proves claim 2.

Claim 3. $\Phi_{\bar{x}}(\hat{x})\le\Phi_{\bar{x}}(0_{H})=\alpha|\bar{x}|_{H}^{2}$, using $|0_{H}|_{V}=0$ and $|0_{H}-\bar{x}|_{H}=|\bar{x}|_{H}$ (Elementary Identities in a Real Inner Product Space (ii), Elementary Identities in a Real Inner Product Space (iv)). If $\bar{x}\in V$, then $\Phi_{\bar{x}}(\hat{x})\le\Phi_{\bar{x}}(\bar{x})=\tfrac12|\bar{x}|_{V}^{2}$; dropping the nonnegative term $\alpha|\hat{x}-\bar{x}|_{H}^{2}$ gives $|\hat{x}|_{V}^{2}\le|\bar{x}|_{V}^{2}$, hence $|\hat{x}|_{V}\le|\bar{x}|_{V}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), and dropping $\tfrac12|\hat{x}|_{V}^{2}$ and multiplying by $\alpha^{-1}>0$ gives $|\hat{x}-\bar{x}|_{H}^{2}\le|\bar{x}|_{V}^{2}/(2\alpha)$.

Claim 4. Let $\hat{y}=R_{\alpha}\bar{y}$. Subtracting the identities (E) for $\bar{x}$ and for $\bar{y}$ and using bilinearity, $\langle\hat{x}-\hat{y},z\rangle_{V}=2\alpha\langle(\bar{x}-\bar{y})-(\hat{x}-\hat{y}),z\rangle_{H}$ for all $z\in V$. With $z=\hat{x}-\hat{y}$ this gives $|\hat{x}-\hat{y}|_{V}^{2}+2\alpha|\hat{x}-\hat{y}|_{H}^{2}=2\alpha\langle\bar{x}-\bar{y},\hat{x}-\hat{y}\rangle_{H}\le 2\alpha|\bar{x}-\bar{y}|_{H}|\hat{x}-\hat{y}|_{H}$ by claim 3 of Properties of the Absolute Value in an Ordered Field and Cauchy-Schwarz. Dropping $|\hat{x}-\hat{y}|_{V}^{2}\ge 0$ and dividing by $2\alpha>0$: $|\hat{x}-\hat{y}|_{H}^{2}\le|\bar{x}-\bar{y}|_{H}|\hat{x}-\hat{y}|_{H}$. If $|\hat{x}-\hat{y}|_{H}=0$ the claim is trivial; otherwise divide by it (claim 5 of Elementary Arithmetic in an Ordered Field with the multiplier $|\hat{x}-\hat{y}|_{H}^{-1}$).

Claim 5. Let $\varepsilon>0$. By condition (b) and Sequential Characterization of the Closure in a Metric Space there is $v\in V$ with $|v-\bar{x}|_{H}<\varepsilon/3$. By claim 3 of The Archimedean Property of the Real Numbers choose $\alpha_{0}>0$ with $|v|_{V}^{2}/(2\alpha_{0})<(\varepsilon/3)^{2}$ (if $|v|_{V}=0$ any $\alpha_{0}>0$ serves). For real $\beta\ge\alpha_{0}$ we have $(2\beta)^{-1}\le(2\alpha_{0})^{-1}$: both inverses are positive (claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field) and multiplying $2\alpha_{0}\le 2\beta$ by the nonnegative number $(2\beta)^{-1}(2\alpha_{0})^{-1}$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives it. Hence claim 3 for $\beta$ gives $|R_{\beta}v-v|_{H}^{2}\le|v|_{V}^{2}/(2\beta)\le|v|_{V}^{2}/(2\alpha_{0})<(\varepsilon/3)^{2}$, so $|R_{\beta}v-v|_{H}<\varepsilon/3$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; and claim 4 for $\beta$ gives $|R_{\beta}\bar{x}-R_{\beta}v|_{H}\le|\bar{x}-v|_{H}<\varepsilon/3$. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (i), $|R_{\beta}\bar{x}-\bar{x}|_{H}\le|R_{\beta}\bar{x}-R_{\beta}v|_{H}+|R_{\beta}v-v|_{H}+|v-\bar{x}|_{H}<\varepsilon$.

Claim 6. Let $\bar{x}\in H$ and put $z_{k}=R_{k}\bar{x}\in D(A)$ for $k\in\mathbb{N}$ (claim 2). Given $\varepsilon>0$, take $\alpha_{0}$ from claim 5 and, by claim 1 of The Archimedean Property of the Real Numbers, $N\in\mathbb{N}$ with $\alpha_{0}<N$ (read in $\mathbb{R}$); for $k\ge N$ we have $k\ge\alpha_{0}$ in $\mathbb{R}$ (claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field when $k\ne N$), so $|z_{k}-\bar{x}|_{H}<\varepsilon$. Thus $(z_{k})$ converges to $\bar{x}$ in $H$, and $\bar{x}$ lies in the closure of $D(A)$ by Sequential Characterization of the Closure in a Metric Space. Hence the closure of $D(A)$ is $H$, which is density.

Claim 7. Let $\bar{x}\in V$ and $z_{k}=R_{k}\bar{x}$. By claim 3 (for $\beta=k$), $|z_{k}|_{V}\le|\bar{x}|_{V}$ for all $k$, and by the argument of claim 6, $(z_{k})$ converges to $\bar{x}$ in $H$. We apply the subsequence criterion The Subsequence Criterion for Convergence in a Metric Space (ii) in $(V,d_{V})$. Let $(z_{k_{j}})_{j}$ be a subsequence. Since $(V,d_{V})$ is separable by hypothesis and $V$ is a real Hilbert space, Bounded Sequences in a Separable Real Hilbert Space Have Weakly Convergent Subsequences applied in $V$ yields a further subsequence $(u_{i})_{i}=(z_{k_{j_{i}}})_{i}$ and $w\in V$ with $u_{i}\rightharpoonup w$ in $V$. By Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator (ii), $u_{i}\rightharpoonup w$ in $H$; but $(u_{i})$ also converges to $\bar{x}$ in $H$ (A Subsequence of a Convergent Sequence Has the Same Limit, twice), hence weakly in $H$ (Elementary Properties of Weak Convergence in a Real Inner Product Space (ii)), so $w=\bar{x}$ by Elementary Properties of Weak Convergence in a Real Inner Product Space (i). Next, the real sequence $(|u_{i}|_{V})_{i}$ converges to $|\bar{x}|_{V}$: otherwise there is $\varepsilon>0$ such that $\bigl||u_{i}|_{V}-|\bar{x}|_{V}\bigr|\ge\varepsilon$ for infinitely many $i$; since $|u_{i}|_{V}\le|\bar{x}|_{V}$, the number $t=|u_{i}|_{V}-|\bar{x}|_{V}$ satisfies $t\le 0$, so $|t|=-t$ (claim 1 of Properties of the Absolute Value in an Ordered Field: $|t|$ is $t$ or $-t$, and $t\le 0\le -t$ by claim 4 of Elementary Order Arithmetic in an Ordered Field), and $\varepsilon\le|t|=-t$ gives $|u_{i}|_{V}\le|\bar{x}|_{V}-\varepsilon$ by sign reversal and translation (claim 4 of Elementary Order Arithmetic in an Ordered Field, claim 3 of Elementary Arithmetic in an Ordered Field); thus these $i$ satisfy $|u_{i}|_{V}\le|\bar{x}|_{V}-\varepsilon$, and they form a strictly increasing sequence of indices (take the least such index above each previous one, by The Natural Numbers Are Well Ordered), giving a subsequence of $(u_{i})$ that still converges weakly to $\bar{x}$ in $V$ (Elementary Properties of Weak Convergence in a Real Inner Product Space (iv)) while being bounded by $|\bar{x}|_{V}-\varepsilon$, so that Elementary Properties of Weak Convergence in a Real Inner Product Space (v) gives $|\bar{x}|_{V}\le|\bar{x}|_{V}-\varepsilon$, a contradiction. By Elementary Properties of Weak Convergence in a Real Inner Product Space (vi) in $V$, $(u_{i})$ converges to $\bar{x}$ in $(V,d_{V})$. Thus every subsequence of $(z_{k})$ has a further subsequence converging to $\bar{x}$ in $V$, and the criterion gives that $(z_{k})$ converges to $\bar{x}$ in $(V,d_{V})$. Since $z_{k}\in D(A)$, Sequential Characterization of the Closure in a Metric Space in $V$ shows that every $\bar{x}\in V$ lies in the closure of $D(A)$ in $(V,d_{V})$; so $D(A)$ is dense in $V$.