What are the approximate eigenvalues of the right shift operator $R$ on $ell_infty$ The Next CEO of Stack OverflowApproximate eigenvalue and continuous spectrumWhat are the Eigenvectors of the curl operator?Spectral theory - continuous spectrumEigenvalues of Left Shift + Right Shift in $l_2([0,infty))$Eigenvalues of the circle over the Laplacian operatorIf an operator has a discrete spectrum eigenvalues, but infinitely many eigenvalues, are these necessarily countable?Spectrum of the right-shift operator on $ell ^2 (mathbbC)$, and a general spectrum questionAre the poles of the resolvent of a Hermitian operator real?The spectrum of a matrix operator is the set of its eigenvalues. But how to prove?Intuitive meaning of the members of the spectrum of a linear operator which are not eigenvalues
What can we do to stop prior company from asking us questions?
Can I equip Skullclamp on a creature I am sacrificing?
What is the point of a new vote on May's deal when the indicative votes suggest she will not win?
Is HostGator storing my password in plaintext?
Is it a good idea to use COLUMN AS (left([Another_Column],(4)) instead of LEFT in the select?
Why did we only see the N-1 starfighters in one film?
Are there languages with no euphemisms?
How should I support this large drywall patch?
Apart from "berlinern", do any other German dialects have a corresponding verb?
Rotate a column
Describing a person. What needs to be mentioned?
What happens if you roll doubles 3 times then land on "Go to jail?"
Why here is plural "We went to the movies last night."
Should I tutor a student who I know has cheated on their homework?
Why do professional authors make "consistency" mistakes? And how to avoid them?
How to use tikz in fbox?
Only print output after finding pattern
What is the purpose of the Evocation wizard's Potent Cantrip feature?
Anatomically Correct Strange Women In Ponds Distributing Swords
MAZDA 3 2006 (UK) - poor acceleration then takes off at 3250 revs
Why were Madagascar and New Zealand discovered so late?
Too much space between section and text in a twocolumn document
How long to clear the 'suck zone' of a turbofan after start is initiated?
How do I construct this japanese bowl?
What are the approximate eigenvalues of the right shift operator $R$ on $ell_infty$
The Next CEO of Stack OverflowApproximate eigenvalue and continuous spectrumWhat are the Eigenvectors of the curl operator?Spectral theory - continuous spectrumEigenvalues of Left Shift + Right Shift in $l_2([0,infty))$Eigenvalues of the circle over the Laplacian operatorIf an operator has a discrete spectrum eigenvalues, but infinitely many eigenvalues, are these necessarily countable?Spectrum of the right-shift operator on $ell ^2 (mathbbC)$, and a general spectrum questionAre the poles of the resolvent of a Hermitian operator real?The spectrum of a matrix operator is the set of its eigenvalues. But how to prove?Intuitive meaning of the members of the spectrum of a linear operator which are not eigenvalues
$begingroup$
I have shown that the spectrum of $R=z$.
Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $forall |z|=1$, $z$ is an approx. eigenvalue. However, are these the only ones?
If they are not how do I find the rest?
Thanks.
functional-analysis operator-theory spectral-theory
edited Mar 18 at 11:59
Andrews
1,2812422
asked Mar 18 at 11:09
Jhon DoeJhon Doe
683414
$endgroup$
add a comment |
$begingroup$
I have shown that the spectrum of $R=z$.
Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $forall |z|=1$, $z$ is an approx. eigenvalue. However, are these the only ones?
If they are not how do I find the rest?
Thanks.
functional-analysis operator-theory spectral-theory
edited Mar 18 at 11:59
Andrews
1,2812422
asked Mar 18 at 11:09
Jhon DoeJhon Doe
683414
$endgroup$
add a comment |
$begingroup$
I have shown that the spectrum of $R=z$.
Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $forall |z|=1$, $z$ is an approx. eigenvalue. However, are these the only ones?
If they are not how do I find the rest?
Thanks.
functional-analysis operator-theory spectral-theory
edited Mar 18 at 11:59
Andrews
1,2812422
asked Mar 18 at 11:09
Jhon DoeJhon Doe
683414
$endgroup$
I have shown that the spectrum of $R=z$.
Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $forall |z|=1$, $z$ is an approx. eigenvalue. However, are these the only ones?
If they are not how do I find the rest?
Thanks.
functional-analysis operator-theory spectral-theory
functional-analysis operator-theory spectral-theory
edited Mar 18 at 11:59
Andrews
1,2812422
asked Mar 18 at 11:09
Jhon DoeJhon Doe
683414
edited Mar 18 at 11:59
Andrews
1,2812422
asked Mar 18 at 11:09
Jhon DoeJhon Doe
683414
edited Mar 18 at 11:59
Andrews
1,2812422
edited Mar 18 at 11:59
Andrews
1,2812422
edited Mar 18 at 11:59
Andrews
1,2812422
1,2812422
asked Mar 18 at 11:09
Jhon DoeJhon Doe
683414
asked Mar 18 at 11:09
Jhon DoeJhon Doe
683414
asked Mar 18 at 11:09
Jhon DoeJhon Doe
683414
683414
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
For all $x in ell_infty$ we have $||Rx||=||x||.$ Thus, for $lambda$ with $|lambda | le 1$ we have
$$(*) ||Rx-lambda x|| ge | quad ||Rx||-|lambda| ||x|| quad|=(1-|lambda|)||x||.$$
Now suppose that $ lambda $ is an approximate eigenvalue . Then there is a sequence $(x_n)$ in $ell_infty$ such that $||x_n||=1$ for all $n$ and $(R-lambda I )x_n to 0$.
From $(*)$ we get: $(1-|lambda|)||x_n|| to 0.$ Since $||x_n||=1$ for all $n$, we derive $|lambda|=1.$
answered Mar 18 at 12:18
FredFred
48.8k11849
$endgroup$
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%2f3152658%2fwhat-are-the-approximate-eigenvalues-of-the-right-shift-operator-r-on-ell-i%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
For all $x in ell_infty$ we have $||Rx||=||x||.$ Thus, for $lambda$ with $|lambda | le 1$ we have
$$(*) ||Rx-lambda x|| ge | quad ||Rx||-|lambda| ||x|| quad|=(1-|lambda|)||x||.$$
Now suppose that $ lambda $ is an approximate eigenvalue . Then there is a sequence $(x_n)$ in $ell_infty$ such that $||x_n||=1$ for all $n$ and $(R-lambda I )x_n to 0$.
From $(*)$ we get: $(1-|lambda|)||x_n|| to 0.$ Since $||x_n||=1$ for all $n$, we derive $|lambda|=1.$
answered Mar 18 at 12:18
FredFred
48.8k11849
$endgroup$
add a comment |
$begingroup$
For all $x in ell_infty$ we have $||Rx||=||x||.$ Thus, for $lambda$ with $|lambda | le 1$ we have
$$(*) ||Rx-lambda x|| ge | quad ||Rx||-|lambda| ||x|| quad|=(1-|lambda|)||x||.$$
Now suppose that $ lambda $ is an approximate eigenvalue . Then there is a sequence $(x_n)$ in $ell_infty$ such that $||x_n||=1$ for all $n$ and $(R-lambda I )x_n to 0$.
From $(*)$ we get: $(1-|lambda|)||x_n|| to 0.$ Since $||x_n||=1$ for all $n$, we derive $|lambda|=1.$
answered Mar 18 at 12:18
FredFred
48.8k11849
$endgroup$
add a comment |
$begingroup$
For all $x in ell_infty$ we have $||Rx||=||x||.$ Thus, for $lambda$ with $|lambda | le 1$ we have
$$(*) ||Rx-lambda x|| ge | quad ||Rx||-|lambda| ||x|| quad|=(1-|lambda|)||x||.$$
Now suppose that $ lambda $ is an approximate eigenvalue . Then there is a sequence $(x_n)$ in $ell_infty$ such that $||x_n||=1$ for all $n$ and $(R-lambda I )x_n to 0$.
From $(*)$ we get: $(1-|lambda|)||x_n|| to 0.$ Since $||x_n||=1$ for all $n$, we derive $|lambda|=1.$
answered Mar 18 at 12:18
FredFred
48.8k11849
$endgroup$
For all $x in ell_infty$ we have $||Rx||=||x||.$ Thus, for $lambda$ with $|lambda | le 1$ we have
$$(*) ||Rx-lambda x|| ge | quad ||Rx||-|lambda| ||x|| quad|=(1-|lambda|)||x||.$$
Now suppose that $ lambda $ is an approximate eigenvalue . Then there is a sequence $(x_n)$ in $ell_infty$ such that $||x_n||=1$ for all $n$ and $(R-lambda I )x_n to 0$.
From $(*)$ we get: $(1-|lambda|)||x_n|| to 0.$ Since $||x_n||=1$ for all $n$, we derive $|lambda|=1.$
answered Mar 18 at 12:18
FredFred
48.8k11849
answered Mar 18 at 12:18
FredFred
48.8k11849
answered Mar 18 at 12:18
FredFred
48.8k11849
answered Mar 18 at 12:18
FredFred
48.8k11849
48.8k11849
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%2f3152658%2fwhat-are-the-approximate-eigenvalues-of-the-right-shift-operator-r-on-ell-i%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
