Definition 1

Let A be a unital algebra over K, and let $e$ be its unit element. For every $x\in A$, the spectrum of $x$ relative to A is the set of $\lambda \in K$ such that $\lambda e-x$ is not invertible.

The spectrum of $x$ will be denoted by Sp$_A(x)$, or Sp($x$) if no confusion can result. The complement of Sp$_A(x)$ in K is called the resolvent set of $x$.