Mixing
Calculate the operator norm of the difference of two operators
Let $x_1,...,x_n in [0,1]$, $lambda_1,...,lambda_n in mathbb{C}$ and let $S,T : C([0,1]) to C([0,1])$ be the operators defined by
$T(g)= int_0^1 g(s), ds$
and
$S(g) = sum_{i=1}^n lambda_i g(x_i)$.
$C([0,1])$ is equipped with the $|cdot|_{infty}$-norm.
I already proved that $|T| = 1$ and $|S| = sum_{i=1}^n |lambda_i|$, where I used for the $geq $-inequality the function $tilde{g}$, which satisfies $tilde{g}(x_i) = dfrac{bar{lambda_i}}{|lambda_i|}$ for all $i=1,...,n$ and is linearly interpolated between the $x_i$.
Now I want to find the norm of the operator $R:=T-S$.
We obviously have
$|R| leq |T|+|S| = 1+sum_{i=1}^n |lambda_i|$
and
$|R| geq ||T|-|S|| = |1-sum_{i=1}^n |lambda_i||$.
How can I proceed now?
Thanks in advance!
functional-analysis operator-theory norm
asked Nov 20 '18 at 14:49
Max93
30529
add a comment |
Let $x_1,...,x_n in [0,1]$, $lambda_1,...,lambda_n in mathbb{C}$ and let $S,T : C([0,1]) to C([0,1])$ be the operators defined by
$T(g)= int_0^1 g(s), ds$
and
$S(g) = sum_{i=1}^n lambda_i g(x_i)$.
$C([0,1])$ is equipped with the $|cdot|_{infty}$-norm.
I already proved that $|T| = 1$ and $|S| = sum_{i=1}^n |lambda_i|$, where I used for the $geq $-inequality the function $tilde{g}$, which satisfies $tilde{g}(x_i) = dfrac{bar{lambda_i}}{|lambda_i|}$ for all $i=1,...,n$ and is linearly interpolated between the $x_i$.
Now I want to find the norm of the operator $R:=T-S$.
We obviously have
$|R| leq |T|+|S| = 1+sum_{i=1}^n |lambda_i|$
and
$|R| geq ||T|-|S|| = |1-sum_{i=1}^n |lambda_i||$.
How can I proceed now?
Thanks in advance!
functional-analysis operator-theory norm
asked Nov 20 '18 at 14:49
Max93
30529
1
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 '18 at 15:01
add a comment |
Let $x_1,...,x_n in [0,1]$, $lambda_1,...,lambda_n in mathbb{C}$ and let $S,T : C([0,1]) to C([0,1])$ be the operators defined by
$T(g)= int_0^1 g(s), ds$
and
$S(g) = sum_{i=1}^n lambda_i g(x_i)$.
$C([0,1])$ is equipped with the $|cdot|_{infty}$-norm.
I already proved that $|T| = 1$ and $|S| = sum_{i=1}^n |lambda_i|$, where I used for the $geq $-inequality the function $tilde{g}$, which satisfies $tilde{g}(x_i) = dfrac{bar{lambda_i}}{|lambda_i|}$ for all $i=1,...,n$ and is linearly interpolated between the $x_i$.
Now I want to find the norm of the operator $R:=T-S$.
We obviously have
$|R| leq |T|+|S| = 1+sum_{i=1}^n |lambda_i|$
and
$|R| geq ||T|-|S|| = |1-sum_{i=1}^n |lambda_i||$.
How can I proceed now?
Thanks in advance!
functional-analysis operator-theory norm
asked Nov 20 '18 at 14:49
Max93
30529
Let $x_1,...,x_n in [0,1]$, $lambda_1,...,lambda_n in mathbb{C}$ and let $S,T : C([0,1]) to C([0,1])$ be the operators defined by
$T(g)= int_0^1 g(s), ds$
and
$S(g) = sum_{i=1}^n lambda_i g(x_i)$.
$C([0,1])$ is equipped with the $|cdot|_{infty}$-norm.
I already proved that $|T| = 1$ and $|S| = sum_{i=1}^n |lambda_i|$, where I used for the $geq $-inequality the function $tilde{g}$, which satisfies $tilde{g}(x_i) = dfrac{bar{lambda_i}}{|lambda_i|}$ for all $i=1,...,n$ and is linearly interpolated between the $x_i$.
Now I want to find the norm of the operator $R:=T-S$.
We obviously have
$|R| leq |T|+|S| = 1+sum_{i=1}^n |lambda_i|$
and
$|R| geq ||T|-|S|| = |1-sum_{i=1}^n |lambda_i||$.
How can I proceed now?
Thanks in advance!
functional-analysis operator-theory norm
functional-analysis operator-theory norm
asked Nov 20 '18 at 14:49
Max93
30529
asked Nov 20 '18 at 14:49
Max93
30529
asked Nov 20 '18 at 14:49
Max93
30529
asked Nov 20 '18 at 14:49
Max93
30529
asked Nov 20 '18 at 14:49
Max93
30529
30529
1
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 '18 at 15:01
add a comment |
1
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 '18 at 15:01
1
1
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 '18 at 15:01
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 '18 at 15:01
add a comment |
0
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',
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%2f3006399%2fcalculate-the-operator-norm-of-the-difference-of-two-operators%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
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%2f3006399%2fcalculate-the-operator-norm-of-the-difference-of-two-operators%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
1
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 '18 at 15:01