Abstract
If the image φ ( A ) \varphi (A) of a normal operator A A on a separable Hilbert space K \mathcal {K} is a diagonal operator for some nonzero representation φ \varphi of B ( K ) B(\mathcal {K}) (that annihilates the compact operators), then A A must itself be a diagonal operator on K \mathcal {K} (with countable spectrum). This yields an “algebraic” characterization of the closure of the range of a derivation induced by a diagonal operator.
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