Abstract
Recent results of M. Radjabalipour and H. Radjavi assert that the sum of a normal operator N with spectrum on a smooth Jordan curve and a compact operator K in the Macaev ideal S ω {\mathfrak {S}_\omega } is decomposable provided the spectrum of N + K N + K does not fill the interior of the curve. Examples are given to show that this result cannot be essentially improved by taking K in a larger ideal.
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