Proof of The Form Operator and the Riesz Map of a Diagonal Hilbert Triple

Each result cited is universally quantified over the data in its own statement. Throughout, $x_{k}=\langle x,e_{k}\rangle_{H}$ and likewise for other vectors of $H$; by Orthonormal Expansions in a Real Hilbert Space (iv) the series $\sum_{k=1}^{\infty}x_{k}^{2}$ converges with sum $|x|_{H}^{2}$ for every $x\in H$, and $\langle v,w\rangle_{H}=\sum_{k=1}^{\infty}v_{k}w_{k}$ for all $v,w\in H$. By claim 1 of Elementary Arithmetic in an Ordered Field and transitivity each $\lambda_{k}$ is positive. By orthonormality, $\langle e_{j},e_{k}\rangle_{H}$ is $1$ if $k=j$ and $0$ otherwise, and by The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights (i) every $e_{j}$ lies in $V$.

Claim 1. Suppose first that $x\in D(A)$ and put $z=Ax$, so that $\langle x,y\rangle_{V}=\langle z,y\rangle_{H}$ for every $y\in V$ by Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iv). Taking $y=e_{j}$ and using The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights (ii) gives

$$\lambda_{j}x_{j}=\langle x,e_{j}\rangle_{V}=\langle z,e_{j}\rangle_{H}=z_{j}\qquad\text{for every }j\in\mathbb{N}.$$

Hence $\lambda_{j}^{2}x_{j}^{2}=z_{j}^{2}$ for every $j$, and the series $\sum_{k=1}^{\infty}z_{k}^{2}$ converges, so $\sum_{k=1}^{\infty}\lambda_{k}^{2}x_{k}^{2}$ converges. Moreover Orthonormal Expansions in a Real Hilbert Space (iii) gives $z=\sum_{k=1}^{\infty}z_{k}e_{k}=\sum_{k=1}^{\infty}\lambda_{k}x_{k}e_{k}$, the two series having the same terms.

Conversely, suppose $x\in H$ and the series $\sum_{k=1}^{\infty}\lambda_{k}^{2}x_{k}^{2}$ converges. Multiplying $1\le\lambda_{k}$ by the nonnegative number $\lambda_{k}x_{k}^{2}$, using claim 5 of Elementary Arithmetic in an Ordered Field, gives $0\le\lambda_{k}x_{k}^{2}\le\lambda_{k}^{2}x_{k}^{2}$ for every $k$, so $\sum_{k=1}^{\infty}\lambda_{k}x_{k}^{2}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (iii) and $x\in V$. By Orthonormal Expansions in a Real Hilbert Space (ii) applied to the coefficients $\lambda_{k}x_{k}$, the series $\sum_{k=1}^{\infty}\lambda_{k}x_{k}e_{k}$ converges in $H$; write $z$ for its sum, so that $z_{j}=\lambda_{j}x_{j}$ for every $j$. For $y\in V$ the series $\sum_{k=1}^{\infty}z_{k}y_{k}$ and $\sum_{k=1}^{\infty}\lambda_{k}x_{k}y_{k}$ have the same terms and converge, the first with sum $\langle z,y\rangle_{H}$ and the second with sum $\langle x,y\rangle_{V}$ by The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights (i). Hence $\langle x,y\rangle_{V}=\langle z,y\rangle_{H}$ for every $y\in V$, so $x\in D(A)$ and $Ax=z$ by Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iv).

Claim 2. Fix $j\in\mathbb{N}$. The terms of $\sum_{k=1}^{\infty}\lambda_{k}^{2}\langle e_{j},e_{k}\rangle_{H}^{2}$ vanish for $k\ne j$ and equal $\lambda_{j}^{2}$ for $k=j$, so by claim 7 of Properties of Finite Sums every partial sum with $j\le n$ equals $\lambda_{j}^{2}$; such a sequence converges to $\lambda_{j}^{2}$, since for a positive $\varepsilon$ every such $n$ gives a difference of absolute value $0<\varepsilon$. So the series converges and $e_{j}\in D(A)$ by claim 1. By claim 1 again, $Ae_{j}$ is the sum of the series $\sum_{k=1}^{\infty}\lambda_{k}\langle e_{j},e_{k}\rangle_{H}e_{k}$, whose terms vanish for $k\ne j$ and equal $\lambda_{j}e_{j}$ for $k=j$; by claim 7 of Properties of Finite Sums of Vectors every partial sum with $j\le n$ equals $\lambda_{j}e_{j}$, and the same argument gives $Ae_{j}=\lambda_{j}e_{j}$.

Claim 3. Let $z\in H$ and put $c_{k}=\tfrac{1}{\lambda_{k}}z_{k}$. Multiplying $1\le\lambda_{k}$ by the positive number $\tfrac{1}{\lambda_{k}}$, using claim 5 of Elementary Arithmetic in an Ordered Field, gives $0<\tfrac{1}{\lambda_{k}}\le1$, whence $(\tfrac{1}{\lambda_{k}})^{2}\le1$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and $0\le c_{k}^{2}\le z_{k}^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field. Since $\sum_{k=1}^{\infty}z_{k}^{2}$ converges, Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (iii) shows that $\sum_{k=1}^{\infty}c_{k}^{2}$ converges, and Orthonormal Expansions in a Real Hilbert Space (ii) shows that the series $\sum_{k=1}^{\infty}c_{k}e_{k}$ converges in $H$; write $x$ for its sum, so that $x_{k}=c_{k}$ for every $k$.

Then $\lambda_{k}^{2}x_{k}^{2}=z_{k}^{2}$ for every $k$, so $\sum_{k=1}^{\infty}\lambda_{k}^{2}x_{k}^{2}$ converges and claim 1 gives $x\in D(A)$ with $Ax=\sum_{k=1}^{\infty}\lambda_{k}x_{k}e_{k}=\sum_{k=1}^{\infty}z_{k}e_{k}=z$, the last equality by Orthonormal Expansions in a Real Hilbert Space (iii). By Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator (iii) the Riesz map satisfies $J(Ax)=x$ for every $x\in D(A)$, so $Jz=J(Ax)=x$, which is the asserted identity.