Indecomposable HilbertSchmidt operators
Author:
Gary Weiss
Journal:
Proc. Amer. Math. Soc. 56 (1976), 172176
DOI:
https://doi.org/10.1090/S00029939197603999160
MathSciNet review:
0399916
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Abstract  References  Additional Information
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.

J. H. Anderson, Derivations, commutators, and the essential numerical range, Dissertation, Indiana University, 1971.
 L. G. Brown, R. G. Douglas, and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of $C^{\ast } $algebras, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Springer, Berlin, 1973, pp. 58–128. Lecture Notes in Math., Vol. 345. MR 0380478
 J. W. Calkin, Twosided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. (2) 42 (1941), 839–873. MR 5790, DOI https://doi.org/10.2307/1968771
 J. William Helton and Roger E. Howe, Integral operators: commutators, traces, index and homology, Proceedings of a Conference Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Springer, Berlin, 1973, pp. 141–209. Lecture Notes in Math., Vol. 345. MR 0390829 Gary Weiss, Commutators and operator ideals, Dissertation, University of Michigan, 1975.
Additional Information
Keywords:
HilbertSchmidt operator,
trace class operator,
<IMG WIDTH="67" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$N + K$">,
decomposable,
ideal,
<!– MATH $2 \times 2$ –> <IMG WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img3.gif" ALT="$2 \times 2$"> operatorvalued matrix,
weighted shift operator
Article copyright:
© Copyright 1976
American Mathematical Society