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!