Let $T$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$, with spectrum $\sigma(T)$. For any $x,y\in \mathcal{H}$, denote by $\mu_{xy}$ the spectral measure of $T$ with respect to $x$ and $y$, that is the unique Borel measure on $\sigma(T)$ such that

$$ \langle x,f(T)y\rangle = \int_{\sigma(T)} f(\lambda)d \mu_{xy}(\lambda) \quad \forall f\in \mathcal{C}(\sigma(T),\mathbb{C}).$$

Then, one can prove that $$ \overline{\bigcup_{x,y\in \mathcal{H}} \text{Supp}(\mu_{xy})} = \sigma(T). $$

Let now $\{e_j\}_{j\in J}$ be an orthonormal basis of $\mathcal{H}$. Then, it is easy to see that $\mu_{e_i e_i}\ge 0$ for all $i$ and that $\mu_{e_i e_j}(\sigma(T))$ equals 1 if $i=j$ and 0 otherwise.

Since $\{e_j\}_{j\in J}$ is an orthonormal basis for $\mathcal{H}$, I wanted to try and prove that it is possible to recover $\sigma(T)$ from the supports of the measures $\{\mu_{e_i e_i}\}_i$. This would be straightforward if the measures $\mu_{e_i e_j}$ were all to be positive, however I don't see why this would be the case. My question is thus: is it true that

$$ \overline{\bigcup_{j\in J} \text{Supp}(\mu_{e_j e_j})} = \sigma(T)? $$