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+---
+title: Test
+parent: Test
+grand_parent: Spectral Theory
+nav_order: 2
+description: >
+  The
+# spellchecker:words Steinhaus preimages Baire pointwise
+---
+
+# {{ page.title }}
+
+{: .definition-title }
+> 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-title }
+> 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.