Definition 8

Let $u$ be a closed operator with dense domain on a complex Banach space E. The resolvent set of $u$ is the set of complex numbers $\lambda$ such that the partial operator $\lambda 1_E-u$ is injective and its inverse $(\lambda 1_E-u)^{-1}$ belongs to $\mathscr{L}(E)$.

The spectrum of $u$, denoted by Sp($u$), is the complement of the resolvent set in $\mathbf{C}$.

If $\lambda \in \mathbf{C}-$ Sp($u$), we denote by $R(u, \lambda )\in \mathscr{L}(E)$ the inverse of $\lambda 1_E-u$. The mapping of $\mathbf{C}-$ Sp($u$) into $\mathscr{L}(E)$ which associates $R(u, \lambda )$ to $\lambda$ is called the resolvent of $u$.