Definition 6.2.1 (Spectrum).label Let $A$ be a unital algebra over $\com$. For each $x\in A$, the set
\begin{align*}\spec{A}{x}=\curl{\lam\in\com:(x-\lam)\text{ is not invertible}}\end{align*}
is called the spectrum of $x$ in $A$. The complement of $\spec{A}{x}$ in $\com$ is called the resolvent of $x$. Note that we identify $\lam\in\com$ with $\lam 1$ in the algebra.
If $A$ is not unital then the quasi-spectrum $\qspec{A}{x}$ of $x\in A$ is the spectrum $\spec{A_I}{x}$ of $x$ in $A_{I}$ where $A_{I}$ is the algebra obtained by adjunction of an identity to $A$. The quasi-spectrum $\qspec{A}{x}$ always contains zero.