summaryrefslogtreecommitdiffstats log msg author committer range
path: root/pages/spectral-theory/test/basic.md
blob: 8c42f6df90f367cc5a376211e200e2bf195f8ec5 (plain)
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59  --- 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.