Assuming the $R_n$ are orthogonal projections the answer is no, in general.

Since orthogonal projections on orthogonal spaces commute, the sum $\tilde{A} = \sum_n \alpha_n R_n$ commutes with its adjoint $\sum_n \alpha_n^* R_n$. Conversely, a bounded normal operator has an orthogonal spectral deposition.

Assuming the $R_n$ are orthogonal projections the answer is no, in general.

Since orthogonal projections on orthogonal subspaces commute, the sum $\tilde{A} = \sum_n \alpha_n R_n$ commutes with its adjoint $\sum_n \alpha_n^* R_n$. Conversely, a bounded normal operator has an orthogonal spectral deposition.