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

In the setting of Hilbert Triples: Standing Notation and Background, let $(H,V,A)$ be the diagonal Hilbert triple determined by an orthonormal basis $(e_{k})_{k\in\mathbb{N}}$ of $H$ and a sequence $(\lambda_{k})_{k\in\mathbb{N}}$ of real numbers with $1\le\lambda_{k}$ for every $k\in\mathbb{N}$; the standing separability hypothesis Hilbert Triples: Standing Notation and Background (ii) holds for this triple by The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights (v). For $x\in H$ write $x_{k}=\langle x,e_{k}\rangle_{H}$. Then the following hold.

1. (The domain and the operator)(i) The domain $D(A)$ is the set of those $x\in H$ for which the series $\sum_{k=1}^{\infty}\lambda_{k}^{2}x_{k}^{2}$ converges. For such $x$ the series $\sum_{k=1}^{\infty}\lambda_{k}x_{k}e_{k}$ converges in $H$ and

$$Ax=\sum_{k=1}^{\infty}\lambda_{k}x_{k}e_{k}.$$

2. (The basis vectors are eigenvectors)(ii) For every $j\in\mathbb{N}$ the vector $e_{j}$ lies in $D(A)$ and

$$Ae_{j}=\lambda_{j}e_{j}.$$

3. (The Riesz map)(iii) For every $z\in H$ the series $\sum_{k=1}^{\infty}\tfrac{1}{\lambda_{k}}z_{k}e_{k}$ converges in $H$ and

$$Jz=\sum_{k=1}^{\infty}\tfrac{1}{\lambda_{k}}z_{k}e_{k}.$$