Recently, I came across the following question while studying Fredholm operator. Recall an operator $S$ on a Hilbert space $\mathcal H$ is said to be Fredholm if $Range(S)$ is closed along with both $ker S$ and $ker(S^*)$ is finite dimensional. My question is as follows:

Let $T$ be a bounded operator on separable complex Hilbert space $\mathcal H$ such that it's spectrum $\sigma(T) \subseteq \overline{\mathbb D}.$ Assume $(T-wI)$ is fredholm operator for every $w\in \mathbb D.$ Does it imply $\varphi(T)-w$ is also fredholm for every $w\in \mathbb D$ and for every automorphism $\varphi$ of the unit disc $\mathbb D?$