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Given an operator $T\in\mathcal{L}(\mathcal{H})$, where $\mathcal{H}$ is a separable Hilbert space, the similarity orbit of $T$ is defined by \begin{equation} SO(T)=\{STS^{-1}:S\in\mathcal{L}(\mathcal{H})\}. \end{equation}

I read about this theory in some papers but I wonder whether there is some good books discussing this issue systematically. I am particularly interested in properties like what is the infimum of norm of operators in $SO(T)$ and how far is the orbit from diagonal operators? compact operators? finite rank operators?

Thanks!

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$\inf\{\|x\|:x\in SO(T)\}$ is the spectral radius of $T$. See math.stackexchange.com/questions/140676/c-algebra-question –  Jonas Meyer Jun 27 '12 at 15:16
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If you have access to MathSciNet, look up Herrero's papers and those which cite them –  user16299 Jun 28 '12 at 6:59
    
If you don't have access to MathSciNet, try this Google Scholar search: scholar.google.com/…. There are several relevant looking articles freely available in pdf format. –  Jonas Meyer Jul 1 '12 at 5:15
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up vote 1 down vote accepted

I really want to close this problem. As mentioned by user6299 in his/her comment, Herrero has done a lot of work on problems related to orbits of operators in Hilbert spaces. Also a good referece (but quite hard) is his book Approximation of Hilbert Space Operators.

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