Application of the closed graph theorem
Let $(mathcal{H},langlecdot,cdotrangle)$ be a Hillbert Space, $T: mathcal{H}tomathcal{H}$ a linear map and
$$rho(T) = {zinmathbb{C}: T-z text{is bijective and} (T-z)^{-1}in mathcal{L}(mathcal{H})}$$
the resolvent set of $T$.
- Suppose $rho(T)neq emptyset.$ Prove that $Tin mathcal{L}(mathcal{H}).$
- Suppose $Tinmathcal{L}(mathcal{H}).$ Prove that $zinrho(T)$ if and only if $T-z$ is bijective.
- Suppose $Tinmathcal{L}(mathcal{H}).$ Prove that $T$ is injective with closed range if and only if there exists a constant $C>0$ such that for any $uinmathcal{H}$ we have $$||Tu||geq C||u||.$$
I think it's not very difficult, but I have some troubles..
- it exists $zinmathbb{C}$ such that $T-z$ is bijective and $(T-z)^{-1}in mathcal{L}(mathcal{H})$. Why can I conclude $Tin mathcal{L}(mathcal{H})?$
$T-z$ is bijective, $T$ is continuous $Rightarrow$ $(T-z)^{-1}$ is continuous $Rightarrow$ $zinrho(T)$
- I think I should use the closed graph theorem here, but unfortunately I have no idea.
Thank you for help!
functional-analysis
asked Nov 28 '18 at 2:09
hAM1t
33
add a comment |
Let $(mathcal{H},langlecdot,cdotrangle)$ be a Hillbert Space, $T: mathcal{H}tomathcal{H}$ a linear map and
$$rho(T) = {zinmathbb{C}: T-z text{is bijective and} (T-z)^{-1}in mathcal{L}(mathcal{H})}$$
the resolvent set of $T$.
- Suppose $rho(T)neq emptyset.$ Prove that $Tin mathcal{L}(mathcal{H}).$
- Suppose $Tinmathcal{L}(mathcal{H}).$ Prove that $zinrho(T)$ if and only if $T-z$ is bijective.
- Suppose $Tinmathcal{L}(mathcal{H}).$ Prove that $T$ is injective with closed range if and only if there exists a constant $C>0$ such that for any $uinmathcal{H}$ we have $$||Tu||geq C||u||.$$
I think it's not very difficult, but I have some troubles..
- it exists $zinmathbb{C}$ such that $T-z$ is bijective and $(T-z)^{-1}in mathcal{L}(mathcal{H})$. Why can I conclude $Tin mathcal{L}(mathcal{H})?$
$T-z$ is bijective, $T$ is continuous $Rightarrow$ $(T-z)^{-1}$ is continuous $Rightarrow$ $zinrho(T)$
- I think I should use the closed graph theorem here, but unfortunately I have no idea.
Thank you for help!
functional-analysis
asked Nov 28 '18 at 2:09
hAM1t
33
add a comment |
Let $(mathcal{H},langlecdot,cdotrangle)$ be a Hillbert Space, $T: mathcal{H}tomathcal{H}$ a linear map and
$$rho(T) = {zinmathbb{C}: T-z text{is bijective and} (T-z)^{-1}in mathcal{L}(mathcal{H})}$$
the resolvent set of $T$.
- Suppose $rho(T)neq emptyset.$ Prove that $Tin mathcal{L}(mathcal{H}).$
- Suppose $Tinmathcal{L}(mathcal{H}).$ Prove that $zinrho(T)$ if and only if $T-z$ is bijective.
- Suppose $Tinmathcal{L}(mathcal{H}).$ Prove that $T$ is injective with closed range if and only if there exists a constant $C>0$ such that for any $uinmathcal{H}$ we have $$||Tu||geq C||u||.$$
I think it's not very difficult, but I have some troubles..
- it exists $zinmathbb{C}$ such that $T-z$ is bijective and $(T-z)^{-1}in mathcal{L}(mathcal{H})$. Why can I conclude $Tin mathcal{L}(mathcal{H})?$
$T-z$ is bijective, $T$ is continuous $Rightarrow$ $(T-z)^{-1}$ is continuous $Rightarrow$ $zinrho(T)$
- I think I should use the closed graph theorem here, but unfortunately I have no idea.
Thank you for help!
functional-analysis
asked Nov 28 '18 at 2:09
hAM1t
33
Let $(mathcal{H},langlecdot,cdotrangle)$ be a Hillbert Space, $T: mathcal{H}tomathcal{H}$ a linear map and
$$rho(T) = {zinmathbb{C}: T-z text{is bijective and} (T-z)^{-1}in mathcal{L}(mathcal{H})}$$
the resolvent set of $T$.
- Suppose $rho(T)neq emptyset.$ Prove that $Tin mathcal{L}(mathcal{H}).$
- Suppose $Tinmathcal{L}(mathcal{H}).$ Prove that $zinrho(T)$ if and only if $T-z$ is bijective.
- Suppose $Tinmathcal{L}(mathcal{H}).$ Prove that $T$ is injective with closed range if and only if there exists a constant $C>0$ such that for any $uinmathcal{H}$ we have $$||Tu||geq C||u||.$$
I think it's not very difficult, but I have some troubles..
- it exists $zinmathbb{C}$ such that $T-z$ is bijective and $(T-z)^{-1}in mathcal{L}(mathcal{H})$. Why can I conclude $Tin mathcal{L}(mathcal{H})?$
$T-z$ is bijective, $T$ is continuous $Rightarrow$ $(T-z)^{-1}$ is continuous $Rightarrow$ $zinrho(T)$
- I think I should use the closed graph theorem here, but unfortunately I have no idea.
Thank you for help!
functional-analysis
functional-analysis
asked Nov 28 '18 at 2:09
hAM1t
33
asked Nov 28 '18 at 2:09
hAM1t
33
asked Nov 28 '18 at 2:09
hAM1t
33
asked Nov 28 '18 at 2:09
hAM1t
33
asked Nov 28 '18 at 2:09
hAM1t
33
33
add a comment |
add a comment |
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