Test

Definition (resolvent operator, regular value, resolvent set, spectrum, spectral value)

Let $T : \dom{T} \to X$ 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 $\ran{T_{\lambda}}$), 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 $T : \dom{T} \to X$ be an operator in a complex normed space $X$. The point spectrum $\pspec{T}$ is the set of all $\lambda \in \CC$ for which the resolvent $R_\lambda(T)$ does not exist. The residual spectrum $\rspec{T}$ is the set of all $\lambda \in \CC$ for which the resolvent $R_\lambda(T)$ exists, but is not densely defined. The continuous spectrum $\cspec{T}$ is the set of all $\lambda \in \CC$ for which the resolvent $R_\lambda(T)$ exists and is densely defined, but unbounded.

If $R_\lambda(T)$ exists,	is densely defined	and is bounded …	… then $\lambda$ belongs to the
✗	-	-	point spectrum $\pspec{T}$
✓	✗	?	residual spectrum $\rspec{T}$
✓	✓	✗	continuous spectrum $\cspec{T}$
✓	✓	✓	resolvent set $\rho(T)$

By definition, the sets $\pspec{T}$, $\rspec{T}$, $\cspec{T}$ and $\rho(T)$ form a partition of the complex plane.