Test

Let $T:D(T)\to X$ be an operator in a complex normed space $X$. We write

$$T_{\lambda }=T-\lambda =T-\lambda I,$$

where $\lambda$ is a complex number and $I$ is the identical operator on the domain of $T$. If the operator $T_{\lambda }$ is injective, that is, it has an inverse $T_{\lambda }^{-1}$ (with domain $R(T_{\lambda })$), then we call

$$R_{\lambda }(T)=T_{\lambda }^{-1}=(T-\lambda {)}^{-1}=(T-\lambda I{)}^{-1}$$

the resolvent operator of $T$ for $\lambda$. A regular value of $T$ is a complex number $\lambda$ for which the resolvent $R_{\lambda }(T)$ exists, has dense domain and is bounded. The set of all regular values of $T$ is called the resolvent set of $T$ and denoted $\rho (T)$. The complement of the resolvent set in the complex plane is called the spectrum of $T$ and denoted $\sigma (T)$. The elements of the spectrum of $T$ are called the spectral values of $T$.

Definition (Point Spectrum, Residual Spectrum, Continuous Spectrum)

Let $T:D(T)\to X$ be an operator in a complex normed space $X$. The point spectrum ${\sigma }_p(T)$ is the set of all $\lambda \in \mathbb{C}$ for which the resolvent $R_{\lambda }(T)$ does not exist. The residual spectrum ${\sigma }_r(T)$ is the set of all $\lambda \in \mathbb{C}$ for which the resolvent $R_{\lambda }(T)$ exists, but is not densely defined. The continuous spectrum ${\sigma }_c(T)$ is the set of all $\lambda \in \mathbb{C}$ for which the resolvent $R_{\lambda }(T)$ exists and is densely defined, but unbounded.

By definition, the sets ${\sigma }_p(T)$, ${\sigma }_r(T)$, ${\sigma }_c(T)$ and $\rho (T)$ form a partition of the complex plane.