On zero-diagonal operators and traces

Authors:
Peng Fan, Che Kao Fong and Domingo A. Herrero

Journal:
Proc. Amer. Math. Soc. **99** (1987), 445-451

MSC:
Primary 47A12; Secondary 47B10

DOI:
https://doi.org/10.1090/S0002-9939-1987-0875378-9

MathSciNet review:
875378

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Abstract: A Hilbert space operator is called zero-diagonal if there exists an orthonormal basis such that for all . It is known that is the norm limit of a sequence of zero-diagonal operators iff , the essential numerical range of . Our first result says that if and is an ideal of compact operators strictly larger than the trace class, then the sequence can be chosen so that ( cannot be replaced by the trace class!). If is zero-diagonal, then the series converges to a value (zero) that can be called "the trace of with respect to the basis ". Our second result provides, for each operator , the structure of the set of all possible "traces" of (in the above sense). In particular, this set is always either the whole complex plane, a straight line, a singleton, or the empty set.

**[1]**J. H. Anderson,*Derivations, commutators and the essential numerical range*, Thesis, Indiana University, 1971.**[2]**A. Ben-Artzi,*Traces of compact operators*, Integral Equations Operator Theory**7**(1984), 310-324. MR**756762 (86g:47018)****[3]**P. Fan,*On the diagonal of an operator*, Trans. Amer. Math. Soc.**283**(1984), 239-251. MR**735419 (86b:47034)****[4]**P. A. Fillmore, J. G. Stampfli, and J. P. Williams,*On the essential numerical range, the essential spectrum and a problem of Halmos*, Acta Sci. Math. (Szeged)**33**(1972), 179-192. MR**0322534 (48:896)****[5]**P. R. Halmos,*Finite-dimensional vector spaces*, Van Nostrand, Princeton, N. J., 1958. MR**0089819 (19:725b)****[6]**-,*A Hilbert space problem book*, Van Nostrand, Princeton, N. J., 1967. MR**0208368 (34:8178)****[7]**D. A. Herrero,*An essay on quasitriangularity*, Proc. 11th Internat. Conf. on Operator Theory, Bucharest 1986 (Romania), Operator Theory: Advances and Applications, Birkhäuser-Verlag, Basel (to appear). MR**942918 (89h:47028)****[8]**T. Kato,*Perturbation theory for linear operators*, Springer-Verlag, New York, 1966. MR**0203473 (34:3324)****[9]**R. Schatten,*Norm ideals of completely continuous operators*, Springer-Verlag, Berlin, 1960. MR**0119112 (22:9878)****[10]**Q. F. Stout,*Schur products of operators and the essential numerical range*, Trans. Amer. Math. Soc.**264**(1981), 39-47. MR**597865 (82h:47029)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0875378-9

Keywords:
Zero-diagonal,
essential numerical range,
trace,
set of traces,
trace class operators

Article copyright:
© Copyright 1987
American Mathematical Society