Indecomposable HilbertSchmidt operators
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 Proc. Amer. Math. Soc. 56 (1976), 172176 Request permission
Abstract:
In 1973, L. G. Brown, R. G. Douglas, and P. A. Fillmore characterized the set of all operators of the form $N + K$ where $N$ is a normal operator and $K$ is a compact operator and they asked whether or not every HilbertSchmidt operator is the sum of a normal operator and a trace class operator. They later asked if, for every HilbertSchmidt operator $A$, there exists a normal operator $N$ for which $A \oplus N$ is the sum of a normal operator and a trace class operator. We produce a large class of HilbertSchmidt operators $A$ none of which is the sum of a normal operator and a trace class operator, and furthermore, for each arbitrary operator $Q,A \oplus Q$ is not the sum of a normal operator and a trace class operator. We then use this to show that their characterization of the operators $N + K$ does not hold true if we replace the class of compact operators by the trace class or by any ideal $I$ for which $I \ne {I^{1/2}}$. In the case of the trace class, we show that even if the vanishing of the Helton and Howe trace invariant were added to the hypothesis of their characterization, it would not hold true.References

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Additional Information
 © Copyright 1976 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 56 (1976), 172176
 DOI: https://doi.org/10.1090/S00029939197603999160
 MathSciNet review: 0399916