Resolvent set of a Banach algebra element
= Resolvent set of a Banach algebra element
{title2=$\rho_A(x)=\mathbb C\setminus\sigma_A(x)$}
= Algebra resolvent set
{synonym}
The <algebra resolvent set> consists of those scalars for which $\lambda1-x$ is invertible in the algebra. It is <open> by the <Neumann series>, and the inverse-valued map is the <resolvent of an element>. A <closed> <subalgebra> can have a smaller <algebra resolvent set> because an ambient inverse need not lie in it.