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Real spectrum, not self-adjoint

  1. Oct 12, 2008 #1
    Let [tex]X[/tex] be a [tex]C^*[/tex]-algebra. I know that if [tex]x\in X[/tex] is self-adjoint, then its spectrum is real, [tex]\sigma(x)\subset\mathbb{R}[/tex]. I haven't seen a claim about the converse, but it seems difficult to come up with a counter example for it. My question is, that is it possible, that some [tex]x\in X[/tex] has a real spectrum, but still [tex]x^*\neq x[/tex]?
  2. jcsd
  3. Oct 12, 2008 #2


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    Yes it is. Take for instance the 2x2 matrix (so [itex]X=M_2(\mathbb{C})[/itex])

    [tex]x = \begin{pmatrix}a & 1 \\ 0 & b\end{pmatrix},[/tex]

    where a and b are any real numbers. The spectrum of x is {a,b} but x is not selfadjoint.
  4. Oct 13, 2008 #3
    I see.

    (hmhmhmh.... I didn't receive mail notification of your response...)
  5. Oct 13, 2008 #4


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    You will only get e-mail notification if you "subscribe" to a thread. To do that, clilck on "Thread Tools" at the top of the thread, then click on "Subscribe to this Thread".
  6. Oct 13, 2008 #5
    I am facing the same problem (see https://www.physicsforums.com/showthread.php?t=257751)

    On the internet I found a reference, however I don't have acces to it:

    In general you can not say anything about the eigenvalues of a real (unsymmetric) matrix. However, if you can write your matrix as a product of matrices then analyzing them you may say something about the eigenvalues of the big matrix.

    I put here two articles, maybe you will find them usefull.

    Attached Files:

    Last edited by a moderator: Apr 23, 2017
  7. Oct 13, 2008 #6
    But isn't the subscribing automatic, so that one has to unsubscribe a thread if one doesn't want notifications. I didn't do anything with thread tools, and I got the notification of your post now.

    There is a non-zero probability for the possibility, that I casually destroyed the first notification without later remembering it. I cannot know it for sure, of course... I was merely mentioning the remark anyway.
  8. Oct 13, 2008 #7


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    When you initially join this forum you are offered the option of automatic "subscription" or not. I chose not because I don't want an e-mail everytime someone responds to one of the threads I responded to. I can't delete all those e-mails AND respond to questions!
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