Let $H$ be  a separable  Hilbert space  and $A$ is  an invertible  bounded operator on $H$. Can we approximate  $A$ with  an invertible operator $B$ such that $sp(B)$ is  a countable set?

**Motivation:**

If the answer is yes, this  would give's us  an alternative  proof  of  connected ness of $GL(H)$. This  alternative proof is identical to a short and interesting proof  of  connectedness  of  $GL(n,\mathbb{C})$, in page  19 of "Introduction to the  Baums  Connes conjecture" by Alain Valette