The exact finite matrix scope

Suppose the $q\times q$ distributions are defined from an orthonormal family $u_1,\dots ,u_q\subset \mathcal{E}$. Vanishing of $W_{raj}$ proves the named matrix elements and, by sesquilinearity, every such matrix element between vectors in their span.

It does not prove that the uncompressed residual vanishes on all of $\mathcal{E}$, or that any closed generator has the expected graph domain.

An entrywise estimate

gives

This is a finite matrix norm estimate, not a field-operator energy bound.