Suppose that $A$ is an bounded linear operator on a Hilbert space such that $\left\|A\right\| \leq 1$. 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 but did not receive any responses. 



Approximation of an Operator
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.


