Eigenvalues are still eigenvalues on restricted subspace?

Question

Let's assume we have some Hilbert space $H$ and for any $k\in \mathbb{N}$ let $H_k\subset H$ be a subspace such that $$\bigoplus_{k\in \mathbb{N}} H_k =H$$ and all $H_k$ are mutually orthogonal. Let $T:\operatorname{dom}(T)\subset H\to H$ be some linear, bounded, closed and densely defined operator. We assume that $T$ leaves $H_k$ invariant for any $k$ and $H_k\subset dom(T)$ for all $k\in \mathbb{N}$. Assume there exists $\lambda \in \sigma(T)$ some eigenvalue of $T$, i.e. $\lambda-T$ is not injective. Can we find some $k_0\in \mathbb{N}$ so that $\lambda$ can be associated with some $x\in H_{k_0}$,i.e. $Tx=\lambda x$? Due to the property of the direct sum, for any $x,x'\in H$ we have $$x=\sum_{k\in K}x_k,\quad x'=\sum_{k'\in K'}x_{k'}$$ for some finite sets $K,K'$. Let $x,x'\in H$, $x\neq x'$ so that $$(\lambda-T)x=(\lambda-T)x'\Leftrightarrow \sum_{k\in K} (\lambda-T)x_k=\sum_{k'\in K'} (\lambda-T)x_{k'}.$$ How can I proceed?