Spectrum in a closed unital subalgebra
= Spectrum in a closed unital subalgebra
If a closed unital subalgebra $A\subseteq B$ contains $x$, then $\sigma_A(x)$ is obtained from $\sigma_B(x)$ by adjoining some bounded connected components of its complement. Membership of $(x-\lambda1)^{-1}$ in $A$ is constant on every connected component of the resolvent set, and it always holds on the unbounded component.