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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.

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    $\begingroup$ Approximate in what topology? And what does "mutually orthogonal" mean in a Banach space? $\endgroup$
    – Nate Eldredge
    Commented Oct 14, 2017 at 3:01
  • $\begingroup$ @NateEldredge with respect to the operator norm in a Hilbert space, corrected thanks. $\endgroup$
    – Mustafa Said
    Commented Oct 14, 2017 at 3:02
  • $\begingroup$ What do you mean by approximate? Of course you can write $A= B +( A-B)$ and then minimize some norm of $A-B$. Is that the question? $\endgroup$
    – lcv
    Commented Oct 15, 2017 at 9:14

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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.

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