Indecomposable Hilbert-Schmidt Operators

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 Hilbert-Schmidt operator is the sum of a normal operator and a trace class operator. They later asked if, for every Hilbert-Schmidt 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 Hilbert-Schmidt 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.