Extensions of the Berger-Shaw theorem

Authors:
Don Hadwin and Eric Nordgren

Journal:
Proc. Amer. Math. Soc. **102** (1988), 517-525

MSC:
Primary 47B10; Secondary 47B20

DOI:
https://doi.org/10.1090/S0002-9939-1988-0928971-X

MathSciNet review:
928971

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Abstract: We show how D. Voiculescu's proof of the Berger-Shaw trace inequality for rationally cyclic nearly hyponormal operators can be presented using only elementary operator-theoretic concepts. In addition we show that if is a hyponormal operator whose essential spectrum has zero area, then the question of whether is trace class depends only on the spectral picture of . We also show how a special case of results of Helton-Howe can be derived from the BDF theory.

**[1]**Constantin Apostol,*Quasitriangularity in Hilbert space*, Indiana Univ. Math. J.**22**(1972/73), 817–825. MR**0326450**, https://doi.org/10.1512/iumj.1973.22.22069**[2]**C. A. Berger and Marion Glazerman Ben-Jacob,*Trace class self-commutators*, Trans. Amer. Math. Soc.**277**(1983), no. 1, 75–91. MR**690041**, https://doi.org/10.1090/S0002-9947-1983-0690041-6**[3]**C. A. Berger and B. I. Shaw,*Selfcommutators of multicyclic hyponormal operators are always trace class*, Bull. Amer. Math. Soc.**79**(1973), 1193–1199, (1974). MR**0374972**, https://doi.org/10.1090/S0002-9904-1973-13375-0**[4]**-,*Hyponormality: its analytic consequences*, preprint.**[5]**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****[6]**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****[7]**J. D. Pincus,*On the trace of commutators in the algebra generated by an operator with trace class self-commutator*, Stony Brook Preprint, 1972.**[8]**Joseph G. Stampfli,*Compact perturbations, normal eigenvalues and a problem of Salinas*, J. London Math. Soc. (2)**9**(1974/75), 165–175. MR**0365196**, https://doi.org/10.1112/jlms/s2-9.1.165**[9]**Dan Voiculescu,*A note on quasitriangularity and trace-class self-commutators*, Acta Sci. Math. (Szeged)**42**(1980), no. 1-2, 195–199. MR**576955****[10 D]**Dan Voiculescu,*Some extensions of quasitriangularity*, Rev. Roumaine Math. Pures Appl.**18**(1973), 1303–1320. MR**0328630****[11]**Dan Voiculescu,*Some results on norm-ideal perturbations of Hilbert space operators*, J. Operator Theory**2**(1979), no. 1, 3–37. MR**553861**

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0928971-X

Article copyright:
© Copyright 1988
American Mathematical Society