If $(pi_λ)_λinmathbb R$ is a family of orthogonal projections, do $λ↦left|pi_λxright|_H^2$ and $λ↦pi_λx$ have the same variation? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Extending the domain of the Dirichlet form associated with a symmetric Markov semigroupLet three signed measures such that $lambda_1botmu$ and $lambda_2botmu$, then $left(lambda_1+lambda_2right)botmu$?Show that an operator is self-adjointShow that the eigenvalues of a compact and self-adjoint bounded linear operator are summableIf $B:X×Y→Z$ is bounded and bilinear, $A:Xto Y$ is nuclear and $(e_n)$ is an ONB, are we able to show $sum_n=1^∞left|B(e_n,Ae_n)right|_Z<∞$?If $(H_λ)_λ≥0$ is a spectral decomposition and $π_λ$ is the orthogonal projection onto $H_λ$, then $t↦π_λ$ is increasing and right-continuousIntegrability with respect to a spectral measureSome questions about the spectral composition of a nonnegative self-adjoint operatorShow that the operator associated to a spectral decomposition on a Hilbert space is self-adjointWhat is the usual definition of the spectral measure for a nonnegative self-adjoint operator on a Hilbert space?Norm of the sum of orthogonal projections
what is the log of the PDF for a Normal Distribution?
Putting class ranking in CV, but against dept guidelines
How do living politicians protect their readily obtainable signatures from misuse?
What is the role of と after a noun when it doesn't appear to count or list anything?
New Order #6: Easter Egg
How does light 'choose' between wave and particle behaviour?
What would you call this weird metallic apparatus that allows you to lift people?
Question about this thing for timpani
What does it mean that physics no longer uses mechanical models to describe phenomena?
Why not use the yoke to control yaw, as well as pitch and roll?
What is a more techy Technical Writer job title that isn't cutesy or confusing?
Tips to organize LaTeX presentations for a semester
What is the difference between a "ranged attack" and a "ranged weapon attack"?
Can humans save crash-landed aliens?
Can two people see the same photon?
How many time has Arya actually used Needle?
Most effective melee weapons for arboreal combat? (pre-gunpowder technology)
Should a wizard buy fine inks every time he want to copy spells into his spellbook?
What initially awakened the Balrog?
Tannaka duality for semisimple groups
How to write capital alpha?
Project Euler #1 in C++
Why is it faster to reheat something than it is to cook it?
Weaponising the Grasp-at-a-Distance spell
If $(pi_λ)_λinmathbb R$ is a family of orthogonal projections, do $λ↦left|pi_λxright|_H^2$ and $λ↦pi_λx$ have the same variation?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Extending the domain of the Dirichlet form associated with a symmetric Markov semigroupLet three signed measures such that $lambda_1botmu$ and $lambda_2botmu$, then $left(lambda_1+lambda_2right)botmu$?Show that an operator is self-adjointShow that the eigenvalues of a compact and self-adjoint bounded linear operator are summableIf $B:X×Y→Z$ is bounded and bilinear, $A:Xto Y$ is nuclear and $(e_n)$ is an ONB, are we able to show $sum_n=1^∞left|B(e_n,Ae_n)right|_Z<∞$?If $(H_λ)_λ≥0$ is a spectral decomposition and $π_λ$ is the orthogonal projection onto $H_λ$, then $t↦π_λ$ is increasing and right-continuousIntegrability with respect to a spectral measureSome questions about the spectral composition of a nonnegative self-adjoint operatorShow that the operator associated to a spectral decomposition on a Hilbert space is self-adjointWhat is the usual definition of the spectral measure for a nonnegative self-adjoint operator on a Hilbert space?Norm of the sum of orthogonal projections
$begingroup$
Let $H$ be a $mathbb R$-Hilbert space and $H_lambda$ be a closed subspace of $H$ for $lambdainmathbb R$. Assume $H_lambdasubseteq H_mu$ for all $lambda,muinmathbb R$ with $lambdalemu$ and let $pi_lambda$ denote the orthogonal projection of $H$ onto $H_lambda$ for $lambdainmathbb R$. Moreover, let $$varrho_x(lambda):=left|pi_lambda xright|_H^2;;;textfor lambdainmathbb R$$ and $$operatorname P_x(lambda):=pi_lambda x;;;textfor lambdainmathbb R$$ for $xin H$.
Fix $xin H$. Are we able to show that the variation of $varrho_x$ and $operatorname P_x$ on any interval coincides?
In order to prove the desired claim, let $kinmathbb N$ and $lambda_0,ldots,lambda_kinmathbb R$ with $lambda_0lecdotslelambda_k$. We need to show that $$A:=sum_i=1^kleft|varrho_x(lambda_i)-varrho_x(lambda_i-1)right|=sum_i=1^kleft|operatorname P_x(lambda_i)-operatorname P_x(lambda_i-1)right|_H=:B.$$ Since $varrho_x$ is nondecreasing, $A=varrho_x(lambda_k)-varrho_x(lambda_0)$. Moreover, we easily see that $$left|operatorname P_x(lambda_i)-operatorname P_x(lambda_i-1)right|_H^2=varrho_x(lambda_i)-varrho_x(lambda_i-1)tag1$$ for all $iinleft1,ldots,kright$.
So, it seems like the claim is wrong, if I'm not missing something. I would at least like show that the variation of $operatorname P_x$ is bounded by the variation of $varrho_x$, i.e. $Ble A$, but that seems to be wrong too.
Remark: The question came up to my mind as I was considering the construction of the spectral measure for self-adjoint operators on a Hilbert space. In the construction of that measure, once is basically integrating against a right-continuous function of bounded variation of the form $operatorname P_x$, but one is usually defining the domain of the integration to be $L^2(varrho_x)$. So, it seems like one is thinking that $varrho_x$ and $operatorname P_x$ (which gives rise to a Lebesgue-Stieltjes vector measure) have the same variation.
functional-analysis operator-theory hilbert-spaces bounded-variation total-variation
asked Mar 6 at 17:11
0xbadf00d0xbadf00d
1,62141534
$endgroup$
add a comment |
$begingroup$
Let $H$ be a $mathbb R$-Hilbert space and $H_lambda$ be a closed subspace of $H$ for $lambdainmathbb R$. Assume $H_lambdasubseteq H_mu$ for all $lambda,muinmathbb R$ with $lambdalemu$ and let $pi_lambda$ denote the orthogonal projection of $H$ onto $H_lambda$ for $lambdainmathbb R$. Moreover, let $$varrho_x(lambda):=left|pi_lambda xright|_H^2;;;textfor lambdainmathbb R$$ and $$operatorname P_x(lambda):=pi_lambda x;;;textfor lambdainmathbb R$$ for $xin H$.
Fix $xin H$. Are we able to show that the variation of $varrho_x$ and $operatorname P_x$ on any interval coincides?
In order to prove the desired claim, let $kinmathbb N$ and $lambda_0,ldots,lambda_kinmathbb R$ with $lambda_0lecdotslelambda_k$. We need to show that $$A:=sum_i=1^kleft|varrho_x(lambda_i)-varrho_x(lambda_i-1)right|=sum_i=1^kleft|operatorname P_x(lambda_i)-operatorname P_x(lambda_i-1)right|_H=:B.$$ Since $varrho_x$ is nondecreasing, $A=varrho_x(lambda_k)-varrho_x(lambda_0)$. Moreover, we easily see that $$left|operatorname P_x(lambda_i)-operatorname P_x(lambda_i-1)right|_H^2=varrho_x(lambda_i)-varrho_x(lambda_i-1)tag1$$ for all $iinleft1,ldots,kright$.
So, it seems like the claim is wrong, if I'm not missing something. I would at least like show that the variation of $operatorname P_x$ is bounded by the variation of $varrho_x$, i.e. $Ble A$, but that seems to be wrong too.
Remark: The question came up to my mind as I was considering the construction of the spectral measure for self-adjoint operators on a Hilbert space. In the construction of that measure, once is basically integrating against a right-continuous function of bounded variation of the form $operatorname P_x$, but one is usually defining the domain of the integration to be $L^2(varrho_x)$. So, it seems like one is thinking that $varrho_x$ and $operatorname P_x$ (which gives rise to a Lebesgue-Stieltjes vector measure) have the same variation.
functional-analysis operator-theory hilbert-spaces bounded-variation total-variation
asked Mar 6 at 17:11
0xbadf00d0xbadf00d
1,62141534
$endgroup$
add a comment |
$begingroup$
Let $H$ be a $mathbb R$-Hilbert space and $H_lambda$ be a closed subspace of $H$ for $lambdainmathbb R$. Assume $H_lambdasubseteq H_mu$ for all $lambda,muinmathbb R$ with $lambdalemu$ and let $pi_lambda$ denote the orthogonal projection of $H$ onto $H_lambda$ for $lambdainmathbb R$. Moreover, let $$varrho_x(lambda):=left|pi_lambda xright|_H^2;;;textfor lambdainmathbb R$$ and $$operatorname P_x(lambda):=pi_lambda x;;;textfor lambdainmathbb R$$ for $xin H$.
Fix $xin H$. Are we able to show that the variation of $varrho_x$ and $operatorname P_x$ on any interval coincides?
In order to prove the desired claim, let $kinmathbb N$ and $lambda_0,ldots,lambda_kinmathbb R$ with $lambda_0lecdotslelambda_k$. We need to show that $$A:=sum_i=1^kleft|varrho_x(lambda_i)-varrho_x(lambda_i-1)right|=sum_i=1^kleft|operatorname P_x(lambda_i)-operatorname P_x(lambda_i-1)right|_H=:B.$$ Since $varrho_x$ is nondecreasing, $A=varrho_x(lambda_k)-varrho_x(lambda_0)$. Moreover, we easily see that $$left|operatorname P_x(lambda_i)-operatorname P_x(lambda_i-1)right|_H^2=varrho_x(lambda_i)-varrho_x(lambda_i-1)tag1$$ for all $iinleft1,ldots,kright$.
So, it seems like the claim is wrong, if I'm not missing something. I would at least like show that the variation of $operatorname P_x$ is bounded by the variation of $varrho_x$, i.e. $Ble A$, but that seems to be wrong too.
Remark: The question came up to my mind as I was considering the construction of the spectral measure for self-adjoint operators on a Hilbert space. In the construction of that measure, once is basically integrating against a right-continuous function of bounded variation of the form $operatorname P_x$, but one is usually defining the domain of the integration to be $L^2(varrho_x)$. So, it seems like one is thinking that $varrho_x$ and $operatorname P_x$ (which gives rise to a Lebesgue-Stieltjes vector measure) have the same variation.
functional-analysis operator-theory hilbert-spaces bounded-variation total-variation
asked Mar 6 at 17:11
0xbadf00d0xbadf00d
1,62141534
$endgroup$
Let $H$ be a $mathbb R$-Hilbert space and $H_lambda$ be a closed subspace of $H$ for $lambdainmathbb R$. Assume $H_lambdasubseteq H_mu$ for all $lambda,muinmathbb R$ with $lambdalemu$ and let $pi_lambda$ denote the orthogonal projection of $H$ onto $H_lambda$ for $lambdainmathbb R$. Moreover, let $$varrho_x(lambda):=left|pi_lambda xright|_H^2;;;textfor lambdainmathbb R$$ and $$operatorname P_x(lambda):=pi_lambda x;;;textfor lambdainmathbb R$$ for $xin H$.
Fix $xin H$. Are we able to show that the variation of $varrho_x$ and $operatorname P_x$ on any interval coincides?
In order to prove the desired claim, let $kinmathbb N$ and $lambda_0,ldots,lambda_kinmathbb R$ with $lambda_0lecdotslelambda_k$. We need to show that $$A:=sum_i=1^kleft|varrho_x(lambda_i)-varrho_x(lambda_i-1)right|=sum_i=1^kleft|operatorname P_x(lambda_i)-operatorname P_x(lambda_i-1)right|_H=:B.$$ Since $varrho_x$ is nondecreasing, $A=varrho_x(lambda_k)-varrho_x(lambda_0)$. Moreover, we easily see that $$left|operatorname P_x(lambda_i)-operatorname P_x(lambda_i-1)right|_H^2=varrho_x(lambda_i)-varrho_x(lambda_i-1)tag1$$ for all $iinleft1,ldots,kright$.
So, it seems like the claim is wrong, if I'm not missing something. I would at least like show that the variation of $operatorname P_x$ is bounded by the variation of $varrho_x$, i.e. $Ble A$, but that seems to be wrong too.
Remark: The question came up to my mind as I was considering the construction of the spectral measure for self-adjoint operators on a Hilbert space. In the construction of that measure, once is basically integrating against a right-continuous function of bounded variation of the form $operatorname P_x$, but one is usually defining the domain of the integration to be $L^2(varrho_x)$. So, it seems like one is thinking that $varrho_x$ and $operatorname P_x$ (which gives rise to a Lebesgue-Stieltjes vector measure) have the same variation.
functional-analysis operator-theory hilbert-spaces bounded-variation total-variation
functional-analysis operator-theory hilbert-spaces bounded-variation total-variation
asked Mar 6 at 17:11
0xbadf00d0xbadf00d
1,62141534
asked Mar 6 at 17:11
0xbadf00d0xbadf00d
1,62141534
edited Apr 2 at 10:55
0xbadf00d
asked Mar 6 at 17:11
0xbadf00d0xbadf00d
1,62141534
asked Mar 6 at 17:11
0xbadf00d0xbadf00d
1,62141534
asked Mar 6 at 17:11
0xbadf00d0xbadf00d
1,62141534
1,62141534
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
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%2f3137835%2fif-pi-%25ce%25bb-%25ce%25bb-in-mathbb-r-is-a-family-of-orthogonal-projections-do-%25ce%25bb%25e2%2586%25a6-left%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.
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%2f3137835%2fif-pi-%25ce%25bb-%25ce%25bb-in-mathbb-r-is-a-family-of-orthogonal-projections-do-%25ce%25bb%25e2%2586%25a6-left%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
