Suppose that $A$ is an bounded linear operator on a Banach space such that $||A|| \leq 1$. Can we approximate $A$ by an operator $\tilde{A}$ such that $\tilde{A} = \sum_{n=1}^N \alpha_n R_n$ where $\alpha_j \in [-1, 1]$ and $R_n$ are mutually orthogonal projections?

I asked this question on Math Stack-exchange a few days ago and did not receive an responses.