# Test

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

$T_{λ}=T−λ=T−λI,$where $λ$ is a complex number and $I$ is the identical operator on the domain of $T$. If the operator $T_{λ}$ is injective, that is, it has an inverse $T_{λ}$ (with domain $R(T_{λ})$), then we call

$R_{λ}(T)=T_{λ}=(T−λ)_{−1}=(T−λI)_{−1}$the

resolvent operatorof $T$ for $λ$. Aregular valueof $T$ is a complex number $λ$ for which the resolvent $R_{λ}(T)$ exists, has dense domain and is bounded. The set of all regular values of $T$ is called theresolvent setof $T$ and denoted $ρ(T)$. The complement of the resolvent set in the complex plane is called thespectrumof $T$ and denoted $σ(T)$. The elements of the spectrum of $T$ are called thespectral valuesof $T$.

Definition (Point Spectrum, Residual Spectrum, Continuous Spectrum)

Let $T:D(T)→X$ be an operator in a complex normed space $X$. The

point spectrum$σ_{p}(T)$ is the set of all $λ∈C$ for which the resolvent $R_{λ}(T)$ does not exist. Theresidual spectrum$σ_{r}(T)$ is the set of all $λ∈C$ for which the resolvent $R_{λ}(T)$ exists, but is not densely defined. Thecontinuous spectrum$σ_{c}(T)$ is the set of all $λ∈C$ for which the resolvent $R_{λ}(T)$ exists and is densely defined, but unbounded.

By definition, the sets $σ_{p}(T)$, $σ_{r}(T)$, $σ_{c}(T)$ and $ρ(T)$ form a partition of the complex plane.