Indecomposable Hilbert-Schmidt operators

Author:
Gary Weiss

Journal:
Proc. Amer. Math. Soc. **56** (1976), 172-176

DOI:
https://doi.org/10.1090/S0002-9939-1976-0399916-0

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 where is a normal operator and 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 , there exists a normal operator for which is the sum of a normal operator and a trace class operator. We produce a large class of Hilbert-Schmidt operators none of which is the sum of a normal operator and a trace class operator, and furthermore, for each arbitrary operator 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 does not hold true if we replace the class of compact operators by the trace class or by any ideal for which . 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.

**[1]**J. H. Anderson,*Derivations, commutators, and the essential numerical range*, Dissertation, Indiana University, 1971.**[2]**L. G. Brown, R. G. Douglas, and P. A. Fillmore,*Unitary equivalence modulo the compact operators and extensions of 𝐶*-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****[3]**J. W. Calkin,*Two-sided ideals and congruences in the ring of bounded operators in Hilbert space*, Ann. of Math. (2)**42**(1941), 839–873. MR**0005790**, https://doi.org/10.2307/1968771**[4]**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****[5]**Gary Weiss,*Commutators and operator ideals*, Dissertation, University of Michigan, 1975.

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0399916-0

Keywords:
Hilbert-Schmidt operator,
trace class operator,
,
decomposable,
ideal,
operator-valued matrix,
weighted shift operator

Article copyright:
© Copyright 1976
American Mathematical Society