Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple

We work in the settings of Real Hilbert Spaces: Standing Notation and Background and Euclidean Space and Real Matrices: Standing Notation, the latter used in the dimension $m$ for a natural number $m$ with $1\le m$.

Let $H$ be a real Hilbert space, with its inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$ as fixed there, and let $\mathrm{Sym}(H)$ be the set of bounded symmetric bilinear forms on $H$, with its norm $\lVert\cdot\rVert$, its order $\preceq$, its sums and scalar multiples and its identity form $I$, as fixed there. Let $\mathcal{S}(m)$ be the set of symmetric real $m\times m$ matrices, with its order $\preceq$, its norm $\lVert\cdot\rVert$ and its identity matrix $I_{m}$, zero matrix $0_{m}$, sums, differences, scalar multiples and matrix-vector products, and let $\mathbb{R}^{m}$ carry its dot product, Euclidean norm $\lVert\cdot\rVert$ and initial segment $[m]$. As in those settings the symbol $\preceq$ serves for the order of $\mathcal{S}(m)$ and for that of $\mathrm{Sym}(H)$, and $\lVert\cdot\rVert$ for the norm of a matrix, the norm of a form and the Euclidean norm alike; the arguments determine which is meant.

Let $e\in H^{m}$ be an $m$-tuple in $H$ that is orthonormal, with components $e_{1},\dots,e_{m}$; in this lemma $e_{i}$ always denotes a component of that tuple, the standard basis vectors of $\mathbb{R}^{m}$, written $e_{i}$ there, not being used. Let $\Lambda:H\to\mathbb{R}^{m}$ and $\Lambda^{\sharp}:\mathbb{R}^{m}\to H$ be the two maps determined by $e$, let $M^{\Lambda}\in\mathrm{Sym}(H)$ be the form carried by the coordinate map for $M\in\mathcal{S}(m)$, and let $\Pi$ and $N$ be the projection form and the tail form of $e$. Then the following hold.

1. (The compression)(i) Let $b\in\mathrm{Sym}(H)$. The real $m\times m$ matrix $b^{\flat}$ with entries

$$(b^{\flat})_{ij}=b(e_{i},e_{j})\qquad(i,j\in[m])$$

belongs to $\mathcal{S}(m)$ and satisfies

$$\zeta\cdot\bigl(b^{\flat}\eta\bigr)=b\bigl(\Lambda^{\sharp}\zeta,\Lambda^{\sharp}\eta\bigr)\qquad\text{for all }\zeta,\eta\in\mathbb{R}^{m}.$$

It is called the compression of $b$ along $e$.

2. (Linearity, norm and order)(ii) For all $b,b'\in\mathrm{Sym}(H)$ and every $\lambda\in\mathbb{R}$,

$$(b+b')^{\flat}=b^{\flat}+(b')^{\flat}, \qquad (\lambda b)^{\flat}=\lambda\,b^{\flat}, \qquad \lVert b^{\flat}\rVert\le\lVert b\rVert ,$$

and $b\preceq b'$ implies $b^{\flat}\preceq(b')^{\flat}$.

3. (The identity, projection and tail forms)(iii) $I^{\flat}=I_{m}$, $\Pi^{\flat}=I_{m}$ and $N^{\flat}=0_{m}$. Consequently

$$(b+t\,N)^{\flat}=b^{\flat}\qquad\text{for every }b\in\mathrm{Sym}(H)\text{ and every }t\in\mathbb{R}.$$

4. (Compression inverts inflation)(iv) $(M^{\Lambda})^{\flat}=M$ for every $M\in\mathcal{S}(m)$.