S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Fri, 6 May 2016 15:53:59 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 9 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Fun in infinite dimensional spacePosted: Tue, 22 Nov 2011 17:10:02 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15248
Location: Austin, TX
Say is a separable Hilbert space and and is strictly positive.

Then what is the minimal value for ?

I had this on a recent exam and I just loved the solution so much, I thought I would share it. Enjoy.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Fun in infinite dimensional spacePosted: Tue, 22 Nov 2011 17:59:59 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7500
Location: NCTS/TPE, Taiwan
Say is a separable Hilbert space and and is strictly positive.

Then what is the minimal value for ?

I had this on a recent exam and I just loved the solution so much, I thought I would share it. Enjoy.

means , the set of bounded linear operators on H? I don't think you need the separability of H.

Spoiler:
Since , by Riesz there exists a unique such that for all . So . Now since T is positive self-adjoint, we get a unique positive self-adjoint square-root , and . So the minimum value is , where is the adjoint to .

_________________

Top

 Post subject: Re: Fun in infinite dimensional spacePosted: Tue, 22 Nov 2011 19:29:08 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15248
Location: Austin, TX
outermeasure wrote:
Say is a separable Hilbert space and and is strictly positive.

Then what is the minimal value for ?

I had this on a recent exam and I just loved the solution so much, I thought I would share it. Enjoy.

means , the set of bounded linear operators on H? I don't think you need the separability of H.

Spoiler:
Since , by Riesz there exists a unique such that for all . So . Now since T is positive self-adjoint, we get a unique positive self-adjoint square-root , and . So the minimum value is , where is the adjoint to .

means compact operators, and though your solution works, there's a particularly cute one when you use that is compact.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Fun in infinite dimensional spacePosted: Wed, 23 Nov 2011 05:28:56 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7500
Location: NCTS/TPE, Taiwan
means compact operators, and though your solution works, there's a particularly cute one when you use that is compact.

Oh, you mean (I think the K comes from German "Kompakter").

It isn't one of those "true for finite rank, so true for all compact" proof, is it?

_________________

Top

 Post subject: Re: Fun in infinite dimensional spacePosted: Wed, 23 Nov 2011 06:00:34 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15248
Location: Austin, TX
outermeasure wrote:
means compact operators, and though your solution works, there's a particularly cute one when you use that is compact.

Oh, you mean , the Calkin algebra on H (I think the K comes from German "Kompakter").

It isn't one of those "true for finite rank, so true for all compact" proof, is it?

No, MUCH cuter.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Fun in infinite dimensional spacePosted: Thu, 24 Nov 2011 19:35:58 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15248
Location: Austin, TX
Here's a hint for those that want one:

Hint 1:
Spoiler:
Try using Hilbert Schmidt

Hint 2:
Spoiler:
By "cute" I mean once you see it in the right light, you can do it using more or less 8th grade information

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Fun in infinite dimensional spacePosted: Sun, 4 Dec 2011 19:52:57 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15248
Location: Austin, TX
And it's been long enough, I'll post the solution, because it's so darn cute.

Spoiler:
By the properties listed for , and since is separable, we have an orthonormal basis of eigenvectors, with associated positive eigenvalues . Now by Riesz, for some fixed .

So we use linearity of and orthonormality of the basis, to write

Now the cute part is that each of these expressions is an upward opening parabola, so the minimum is achieved -- by the quadratic formula -- at , so the minimum occurs at .

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Fun in infinite dimensional spacePosted: Sun, 4 Dec 2011 20:41:50 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15248
Location: Austin, TX
A fun follow-up question I think:

Why is it important that we have and not just ?

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Fun in infinite dimensional spacePosted: Sun, 4 Dec 2011 21:46:58 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7500
Location: NCTS/TPE, Taiwan
And it's been long enough, I'll post the solution, because it's so darn cute.

Spoiler:
By the properties listed for , and since is separable, we have an orthonormal basis of eigenvectors, with associated positive eigenvalues . Now by Riesz, for some fixed .

So we use linearity of and orthonormality of the basis, to write

Now the cute part is that each of these expressions is an upward opening parabola, so the minimum is achieved -- by the quadratic formula -- at , so the minimum occurs at .

I was thinking of how to avoid Riesz and to come up with a natural appearing one, but couldn't find any.

A fun follow-up question I think:

Why is it important that we have and not just ?

Spoiler:
Because you don't get a minimum that way. For example, , and . The resulting supposed minimum would need minimum, i.e. for all n, and that is not in (and the infimum would be ).

_________________

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 9 posts ]

 All times are UTC [ DST ]

Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix and Linear Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous
Privacy Statement | Search the "old" CyberBoard

users online during the last hour