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SPECTRUM, NUMERICAL RANGE AND DAVIS-WIELANDT SHELL OF A NORMAL OPERATOR

Published online by Cambridge University Press:  01 January 2009

CHI-KWONG LI  and
YIU-TUNG POON
CHI-KWONG LI
Affiliation:
Department of Mathematics, College of William & Mary, Williamsburg, VA 23185 e-mail: ckli@math.wm.edu
YIU-TUNG POON
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA 50011 e-mail: ytpoon@iastate.edu
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Abstract

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We denote the numerical range of the normal operator T by W(T). A characterization is given to the points in W(T) that lie on the boundary. The collection of such boundary points together with the interior of the the convex hull of the spectrum of T will then be the set W(T). Moreover, it is shown that such boundary points reveal a lot of information about the normal operator. For instance, such a boundary point always associates with an invariant (reducing) subspace of the normal operator. It follows that a normal operator acting on a separable Hilbert space cannot have a closed strictly convex set as its numerical range. Similar results are obtained for the Davis-Wielandt shell of a normal operator. One can deduce additional information of the normal operator by studying the boundary of its Davis-Wielandt shell. Further extension of the result to the joint numerical range of commuting operators is discussed.

Keywords

47A1047A1247B15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 1 , January 2009 , pp. 91 - 100
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Berberian, S. K., Lectures in functional analysis and operator theory (Springer-Verlag, New York, 1974).CrossRefGoogle Scholar
2.Cho, M., Joint spectra of operators on Banach space, Glasgow Math. J. 28 (1986), 69–72.CrossRefGoogle Scholar
3.Davis, C., The shell of a Hilbert-space operator, Acta Sci. Math.(Szeged) 29 (1968), 69–86.Google Scholar
4.Davis, C., The shell of a Hilbert-space operator. II, Acta Sci. Math. (Szeged) 31 (1970), 301–318.Google Scholar
5.Gustafson, K. E. and Rao, D. K. M., Numerical ranges: The field of values of linear operators and matrices, (Springer, New York, 1997).CrossRefGoogle Scholar
6.Halmos, P., A Hilbert space problem book, Second edition, Graduate Texts in Mathematics, 19, Encyclopedia of Mathematics and its Applications (Spring-Verlag, New York, 1982).CrossRefGoogle Scholar
7.Horn, R. A. and Johnson, C. R., Topics in matrix analysis (Cambridge University Press, Cambridge, 1991).Google Scholar
8.Juneja, P., On extreme points of the joint numerical range of commuting normal operators, Pacific J. Math. 67 (1976), 473–476.CrossRefGoogle Scholar
9.Li, C. K. and Poon, Y. T., Davis-Wielandt shells of normal operators, Acta Sci. Math. (Szeged), in press, preprint available at http://www.resnet.wm.edu/~cklixx/dwshellb.pdf.Google Scholar
10.Wielandt, H., On eigenvalues of sums of normal matrices, Pacific J. Math. 5 (1955), 633–638.CrossRefGoogle Scholar
11.Wrobel, V., Joint spectra and joint numerical ranges for pairwise commuting operators in Banach spaces, Glasgow Math. J. 30 (1988), 145–153.CrossRefGoogle Scholar