Timeline for spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
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| Dec 31, 2012 at 4:25 | comment | added | Alireza Abdollahi | Does anyone check the case in which the subalgebra generated by $a$ and $b$ is nilpotent in the following sense: I mean an algebra $L$ nilpotent if there exists a positive integer $n$ such that the itrated commutator $[L,\dots,L]=0$ (the number of $L$ is $n$), where $[L,L]=\langle ab-ba | a,b\in L\rangle$, $[L,\dots,L]=[[L,\dots,L],L]=\langle ab-ba | a\in [L,\dots,L], b\in L\langle$. The least $n$ is called the nilpotency index of $L$. The main point I think is that the simultaneously diagonal inability often holds for two operators for which the structure generated by them is somehow | |
| Mar 28, 2010 at 7:01 | history | edited | Jonas Meyer |
edited tags
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| Feb 11, 2010 at 0:05 | comment | added | Jonas Meyer | @Nothingwqy: Has Yemon not answered your question? If he has, you can "accept", and if he hasn't, will you please further explain what you're looking for? | |
| Feb 5, 2010 at 9:18 | history | edited | Yemon Choi | CC BY-SA 2.5 |
edited title to be a bit more descriptive/appealing
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| Feb 5, 2010 at 8:43 | comment | added | Yemon Choi | @Jonas: yes, I just thought of that too. I'm working on the extended version right now... | |
| Feb 5, 2010 at 8:32 | comment | added | Jonas Meyer | Oh, the answer to the "more interesting question" is no: Upper triangular matrices. | |
| Feb 5, 2010 at 8:22 | comment | added | Yemon Choi | @Jonas: good point. I don't have a definitive answer but will amend my answer below to say a bit more about this | |
| Feb 5, 2010 at 8:03 | comment | added | Jonas Meyer | @Yemon: That is interesting, but to me it isn't clear exactly what the question above is. I mean, your answer covers it as far as I can tell, but a more interesting question would be the one (I think) you just raised: If these properties of the spectrum are satisfied for all a and b in a Banach algebra A, must A be commutative? | |
| Feb 5, 2010 at 7:59 | history | edited | Yemon Choi |
changed to more relevant tags
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| Feb 5, 2010 at 7:59 | comment | added | Yemon Choi | @Jonas: sorry, I spent quite a while typing and didn't see your comment - hence the duplicate effort below. Regarding your 2nd comment: I think that in the earlier years of the subject, there was some work on showing that conditions like subadditivity and submultiplicativity of spectral radius come close to characterizing commutativity of a Banach algebra. I'd have to look this up, but my gut feeling is that in NC settings one has to give up hope of similar results | |
| Feb 5, 2010 at 7:56 | answer | added | Yemon Choi | timeline score: 4 | |
| Feb 5, 2010 at 7:48 | comment | added | Jonas Meyer | They typically won't hold, but I don't know what you're looking for in an answer. | |
| Feb 5, 2010 at 7:41 | comment | added | Jonas Meyer | For sums you may be interested in the following: mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums/…. For products, consider the case where a is a nonzero nilpotent matrix and b is its conjugate transpose. | |
| Feb 5, 2010 at 7:32 | history | asked | Nothingwqy | CC BY-SA 2.5 |