Application of the closed graph theorem
up vote
0
down vote
favorite
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 at 2:09
hAM1t
33
add a comment |
up vote
0
down vote
favorite
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 at 2:09
hAM1t
33
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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 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 at 2:09
hAM1t
33
asked Nov 28 at 2:09
hAM1t
33
asked Nov 28 at 2:09
hAM1t
33
asked Nov 28 at 2:09
hAM1t
33
asked Nov 28 at 2:09
hAM1t
33
33
add a comment |
add a comment |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3016607%2fapplication-of-the-closed-graph-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3016607%2fapplication-of-the-closed-graph-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
