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Diffstat (limited to 'pages/spectral-theory')
| -rw-r--r-- | pages/spectral-theory/test/basic.md | 32 |
1 files changed, 13 insertions, 19 deletions
diff --git a/pages/spectral-theory/test/basic.md b/pages/spectral-theory/test/basic.md index 05405f4..b1015d1 100644 --- a/pages/spectral-theory/test/basic.md +++ b/pages/spectral-theory/test/basic.md @@ -32,25 +32,19 @@ nav_order: 2 > 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 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 $\rho(T)$ | +{% 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. +{% enddefinition %} -By definition, the sets $\pspec{T}$, $\rspec{T}$, $\cspec{T}$ and $\rho(T)$ form a partition of the complex plane. +By definition, the sets $\pspec{T}$, $\rspec{T}$, $\cspec{T}$ and $\rho(T)$ +form a partition of the complex plane. |