If $B$ is the restriction of an operator $A$ to an invariant closed subspace, is there an inclusion relation between $\rho(A)$ and $\rho(B)$?

Question

Let $A$ be a linear operator on a $\mathbb R$-Banach space $E$ and $K$ be a closed subspace of $E$ such that $A$ is $K$-invariant, i.e. $$A(\mathcal D(A)\cap K)\subseteq K.\tag1$$

Under the given assumptions, $B:=\left.A\right|_{\mathcal D(A)\:\cap\:K}$ is a linear operator on the $\mathbb R$-Banach space $K$. How are the spectra $\sigma(A)$ and $\sigma(B)$ and resolvent sets $\rho(A)$ and $\rho(B)$ of $A$ and $B$ related? Can we show inclusion relations between them?