Elementary Properties of the Trace of a Form along a Square-Summable Sequence

In the setting of Hilbert Triples: Standing Notation and Background, let $\mathrm{Sym}(V)$ and $\mathrm{Sym}(H)$ with their norms, their orders $\preceq$, their sums and scalar multiples, their zero forms $0_{\mathrm{Sym}}$, the identity forms $I_{V}$ and $I_{H}$, and the restriction map $Y\mapsto Y|_{V}$ be as fixed in Hilbert Triples: Standing Notation and Background (vi); the norm of $\mathrm{Sym}(V)$ is written $\lVert\cdot\rVert_{\mathrm{Sym}(V)}$ and that of $\mathrm{Sym}(H)$ simply $\lVert\cdot\rVert$. Let $\mathbb{N}$ be the set of natural numbers, let $f=(f_{k})_{k\in\mathbb{N}}$ be square-summable in $V$, with sum $\sigma(f)$, and let $\mathrm{Tr}_{f}$ be the trace along $f$. We write $|z|_{H}^{2}=|z|_{H}|z|_{H}$ and $|z|_{V}^{2}=|z|_{V}|z|_{V}$. Then the following hold, for all $X,X'\in\mathrm{Sym}(V)$ and every $\lambda\in\mathbb{R}$.

1. (Linearity)(i)

$$\mathrm{Tr}_{f}(X+X')=\mathrm{Tr}_{f}X+\mathrm{Tr}_{f}X', \qquad \mathrm{Tr}_{f}(\lambda X)=\lambda\,\mathrm{Tr}_{f}X, \qquad \mathrm{Tr}_{f}0_{\mathrm{Sym}}=0 .$$

2. (Monotonicity)(ii) If $X\preceq X'$, then $\mathrm{Tr}_{f}X\le\mathrm{Tr}_{f}X'$.

3. (Norm bound)(iii) $\bigl|\mathrm{Tr}_{f}X\bigr|\le\lVert X\rVert_{\mathrm{Sym}(V)}\,\sigma(f)$.

4. (The identity form)(iv) $\mathrm{Tr}_{f}I_{V}=\sigma(f)$.

5. (Forms on $H$ restricted to $V$)(v) The series $\sum_{k=1}^{\infty}|f_{k}|_{H}^{2}$ converges; write $\sigma_{H}(f)$ for its sum, so that $0\le\sigma_{H}(f)\le\sigma(f)$. Every $Y\in\mathrm{Sym}(H)$ has $Y|_{V}\in\mathrm{Sym}(V)$ and satisfies

$$\bigl|\mathrm{Tr}_{f}(Y|_{V})\bigr|\le\lVert Y\rVert\,\sigma_{H}(f)\le\lVert Y\rVert\,\sigma(f).$$

6. (The tail forms of an orthonormal basis)(vi) Let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$ with $e_{k}\in V$ for every $k\in\mathbb{N}$, and let $(N_{m})_{m\in\mathbb{N}}$ be its sequence of tail forms. Then $0\le\mathrm{Tr}_{f}(N_{m}|_{V})$ for every $m\in\mathbb{N}$, and the sequence $\bigl(\mathrm{Tr}_{f}(N_{m}|_{V})\bigr)_{m\in\mathbb{N}}$ converges to $0$.