The closure of a similarity orbit is always arcwise connected (Q1110788)

For a bounded linear operator T on a complex, separable Hilbert space H, the similarity orbit of T is the set \(S(T)=\{WTW^{-1}:W\) invertible on \(H\}\). The author proves the result in the title, more precisely, if \(A\in S(T)^-\) (norm closure), there is a continuous function \(\gamma:[0,1]\to S(T)^-\) such that \(\gamma (0)=T\) and \(\gamma (1)=A\). Some open problems and conjectures as to their outcome are given too.