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Induced norms, states, and numerical ranges

Author(s): Chi-Kwong Li; Edward Poon; Hans Schneider
Journal: Proc. Amer. Math. Soc. 132 (2004), 1501-1506.
MSC (2000): Primary 47A12, 47A30, 15A60
Posted: September 18, 2003
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Abstract: It is shown that two induced norms are the same if and only if the corresponding norm numerical ranges or radii are the same, which in turn is equivalent to the vector states and mixed states arising from the norms being the same. The proofs depend on an auxiliary result of independent interest which concerns when two closed convex sets in a topological vector space are multiples of each other.

References:

1.: F. L. Bauer, On the field of values subordinate to a norm, Numer. Math. 4 (1962), 103-113. MR 26:2860
2.: F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Mathematical Society Lecture Note Series 2, Cambridge University Press, Cambridge, 1971. MR 44:5779
3.: R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. MR 92e:15003
4.: N. Nirschl and H. Schneider, The Bauer fields of values of a matrix, Numer. Math. 6 (1964), 355-365. MR 31:871
5.: R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. MR 43:445
6.: H. Schneider and W. G. Strang, Comparison theorems for supremum norms, Numer. Math. 4 (1962), 15-20. MR 24:A1917

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Additional Information:

Chi-Kwong Li
Affiliation: Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795
Email: ckli@math.wm.edu

Edward Poon
Affiliation: Department of Mathematics, Embry-Riddle Aeronautical University, 3200 Willow Creek Road, Prescott, Arizona 86301
Email: edward.poon@erau.edu

Hans Schneider
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: hans@math.wisc.edu

DOI: 10.1090/S0002-9939-03-07213-7
PII: S 0002-9939(03)07213-7
Keywords: Induced norms, vector states, mixed states, norm numerical ranges
Received by editor(s): December 9, 2002
Received by editor(s) in revised form: January 22, 2003
Posted: September 18, 2003
Additional Notes: Research partially supported by an NSF grant
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2003, American Mathematical Society


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