Let $T$ be a self-adjoint operator on a Hilbert space $H$, with spectrum $\sigma(T)$. For any $x∈H$, denote by $μ_x$ the spectral measure of $T$ at $x$, that is the unique Borel measure on $\sigma(T)$ such that

$$ ⟨x,f(T)x⟩ = \int_{\sigma(T)} f(λ) d \mu_{x}(λ) \quad \forall f \in \mathcal{C}(\sigma(T),\mathbb{C}). $$

Then, one can prove that $$ \overline{\bigcup_{x\in X} Supp(\mu_x)}=\sigma(T)$$ for any orthonormal basis $X$ of $H$.

Suppose that there exists a closed subspace $H_0$ of $H$ and a positive integer $k$ such that $$ H=H_0 \overset{\perp}{\oplus} TH_0 \overset{\perp}{\oplus} \dots \overset{\perp}{\oplus} T^{k-1} H_0 \quad \text{and} \quad T^{k}H_0=H_0,$$ and let $X_0$ be an orthonormal basis for $H_0$.

Suppose that $\mu_x(\{0\})>0$ for all $x\in X_0$.

Then, one has $0\in \sigma(T)$. Moreover, since $T^k|_{H_0}$ is surjective, it follows that $\lambda=0$ is an eigenvalue of $T$. Under some additional assumptions I can actually show that $\lambda=0$ must be an isolated eigenvalue. However, I was actually wondering whether that might always be the case.