For a bounded linear operator $A$ on a separable Hilbert space, the similarity orbit of $A$ is the set $S(A)=\{WAW^{-1}: W \text{ is invertible}\}$. I am wondering that if the identity operator $I$ is in the closure of $S(A)$, what can we say about $A$?
I looked here References on similarity orbits of operators, which lead me to the article "The Closure of the Similarity Orbit an a Hilbert Space Operator" by Apostol, Herrero and Voiculescu. In the article they classify $\overline{S(A)}$, but it is very complicated, and I was hoping that with a seemingly simpler question asking
for which $A$ is $I\in \overline{S(A)}$
would be more tractable. I understand that this is a not so specific question, so to perhaps provide more detail, I am interested in how large the set of operators $\{A: I\in \overline{S(A)}\}$ is. Is it norm dense? SOT dense? etc.
Any examples or advice is appreciated. Thank you.