Due to the negative answer to my last question I want to know if at least the following is true:

Let H be an infinite dimensional separable complex Hilbert space with $S^1$-action. Let $\text{Gl}_{S^1}(H)$ be the space of invertible, bounded and equivariant linear maps (from H to H).

Is $\text{Gl}_{S^1}(H)$ path-connected? If not, what is known about the components?

Of course, I would be happy enough if you point out references.