Approximate point spectrum of multiplication operator in $L^p$
$begingroup$
Let $M_phi f=phi f, ,, phiin L^infty(X,Omega,mu),,, fin L^p(X,Omega,mu),, 1leqslant p leqslant infty.$
I finded $$sigma(M_phi)=left{ lambda in mathbb{C} mid not exists epsilon>0 colon |lambda-f(x)|geqslant epsilon ,,text{a.e.} right}$$ and $$sigma_p(M_phi)={ lambda in mathbb{C} colon muleft( {xin X colon f(x)=lambda } right)>0 }.$$
$sigma_{ap}(M_phi)=,?$
functional-analysis spectral-theory
asked Dec 16 '18 at 22:31
QuantumDKQuantumDK
64
$endgroup$
add a comment |
$begingroup$
Let $M_phi f=phi f, ,, phiin L^infty(X,Omega,mu),,, fin L^p(X,Omega,mu),, 1leqslant p leqslant infty.$
I finded $$sigma(M_phi)=left{ lambda in mathbb{C} mid not exists epsilon>0 colon |lambda-f(x)|geqslant epsilon ,,text{a.e.} right}$$ and $$sigma_p(M_phi)={ lambda in mathbb{C} colon muleft( {xin X colon f(x)=lambda } right)>0 }.$$
$sigma_{ap}(M_phi)=,?$
functional-analysis spectral-theory
asked Dec 16 '18 at 22:31
QuantumDKQuantumDK
64
$endgroup$
add a comment |
$begingroup$
Let $M_phi f=phi f, ,, phiin L^infty(X,Omega,mu),,, fin L^p(X,Omega,mu),, 1leqslant p leqslant infty.$
I finded $$sigma(M_phi)=left{ lambda in mathbb{C} mid not exists epsilon>0 colon |lambda-f(x)|geqslant epsilon ,,text{a.e.} right}$$ and $$sigma_p(M_phi)={ lambda in mathbb{C} colon muleft( {xin X colon f(x)=lambda } right)>0 }.$$
$sigma_{ap}(M_phi)=,?$
functional-analysis spectral-theory
asked Dec 16 '18 at 22:31
QuantumDKQuantumDK
64
$endgroup$
Let $M_phi f=phi f, ,, phiin L^infty(X,Omega,mu),,, fin L^p(X,Omega,mu),, 1leqslant p leqslant infty.$
I finded $$sigma(M_phi)=left{ lambda in mathbb{C} mid not exists epsilon>0 colon |lambda-f(x)|geqslant epsilon ,,text{a.e.} right}$$ and $$sigma_p(M_phi)={ lambda in mathbb{C} colon muleft( {xin X colon f(x)=lambda } right)>0 }.$$
$sigma_{ap}(M_phi)=,?$
functional-analysis spectral-theory
functional-analysis spectral-theory
asked Dec 16 '18 at 22:31
QuantumDKQuantumDK
64
asked Dec 16 '18 at 22:31
QuantumDKQuantumDK
64
asked Dec 16 '18 at 22:31
QuantumDKQuantumDK
64
asked Dec 16 '18 at 22:31
QuantumDKQuantumDK
64
asked Dec 16 '18 at 22:31
QuantumDKQuantumDK
64
64
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
$M_phi$ is normal $Rightarrow sigma_{ap}=sigma$
answered Dec 16 '18 at 22:54
QuantumDKQuantumDK
64
$endgroup$
add a comment |
$begingroup$
$$
sigma_{ap}(M_phi) = { lambdainmathbb{C} : mu{ nu : 0 <|phi(nu)-lambda| < epsilon} >0;; forall epsilon > 0 }.
$$
answered Jan 16 at 6:17
DisintegratingByPartsDisintegratingByParts
59.3k42580
$endgroup$
add a comment |
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',
autoActivateHeartbeat: false,
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%2f3043274%2fapproximate-point-spectrum-of-multiplication-operator-in-lp%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$M_phi$ is normal $Rightarrow sigma_{ap}=sigma$
answered Dec 16 '18 at 22:54
QuantumDKQuantumDK
64
$endgroup$
add a comment |
$begingroup$
$M_phi$ is normal $Rightarrow sigma_{ap}=sigma$
answered Dec 16 '18 at 22:54
QuantumDKQuantumDK
64
$endgroup$
add a comment |
$begingroup$
$M_phi$ is normal $Rightarrow sigma_{ap}=sigma$
answered Dec 16 '18 at 22:54
QuantumDKQuantumDK
64
$endgroup$
$M_phi$ is normal $Rightarrow sigma_{ap}=sigma$
answered Dec 16 '18 at 22:54
QuantumDKQuantumDK
64
answered Dec 16 '18 at 22:54
QuantumDKQuantumDK
64
answered Dec 16 '18 at 22:54
QuantumDKQuantumDK
64
answered Dec 16 '18 at 22:54
QuantumDKQuantumDK
64
64
add a comment |
add a comment |
$begingroup$
$$
sigma_{ap}(M_phi) = { lambdainmathbb{C} : mu{ nu : 0 <|phi(nu)-lambda| < epsilon} >0;; forall epsilon > 0 }.
$$
answered Jan 16 at 6:17
DisintegratingByPartsDisintegratingByParts
59.3k42580
$endgroup$
add a comment |
$begingroup$
$$
sigma_{ap}(M_phi) = { lambdainmathbb{C} : mu{ nu : 0 <|phi(nu)-lambda| < epsilon} >0;; forall epsilon > 0 }.
$$
answered Jan 16 at 6:17
DisintegratingByPartsDisintegratingByParts
59.3k42580
$endgroup$
add a comment |
$begingroup$
$$
sigma_{ap}(M_phi) = { lambdainmathbb{C} : mu{ nu : 0 <|phi(nu)-lambda| < epsilon} >0;; forall epsilon > 0 }.
$$
answered Jan 16 at 6:17
DisintegratingByPartsDisintegratingByParts
59.3k42580
$endgroup$
$$
sigma_{ap}(M_phi) = { lambdainmathbb{C} : mu{ nu : 0 <|phi(nu)-lambda| < epsilon} >0;; forall epsilon > 0 }.
$$
answered Jan 16 at 6:17
DisintegratingByPartsDisintegratingByParts
59.3k42580
answered Jan 16 at 6:17
DisintegratingByPartsDisintegratingByParts
59.3k42580
answered Jan 16 at 6:17
DisintegratingByPartsDisintegratingByParts
59.3k42580
answered Jan 16 at 6:17
DisintegratingByPartsDisintegratingByParts
59.3k42580
59.3k42580
add a comment |
add a comment |
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.
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%2f3043274%2fapproximate-point-spectrum-of-multiplication-operator-in-lp%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
