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

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