Induced norms, states, and numerical ranges

Authors:
Chi-Kwong Li, Edward Poon and Hans Schneider

Journal:
Proc. Amer. Math. Soc. **132** (2004), 1501-1506

MSC (2000):
Primary 47A12, 47A30, 15A60

Published electronically:
September 18, 2003

MathSciNet review:
2053358

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Abstract | References | Similar Articles | Additional Information

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.

**1.**F. L. Bauer,*On the field of values subordinate to a norm*, Numer. Math.**4**(1962), 103–113. MR**0145329****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, vol. 2, Cambridge University Press, London-New York, 1971. MR**0288583****3.**Roger A. Horn and Charles R. Johnson,*Topics in matrix analysis*, Cambridge University Press, Cambridge, 1991. MR**1091716****4.**Nicholas Nirschl and Hans Schneider,*The Bauer fields of values of a matrix*, Numer. Math.**6**(1964), 355–365. MR**0176599****5.**R. Tyrrell Rockafellar,*Convex analysis*, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR**0274683****6.**Hans Schneider and W. Gilbert Strang,*Comparision theorems for supremum norms*, Numer. Math.**4**(1962), 15–20. MR**0132070**

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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:
http://dx.doi.org/10.1090/S0002-9939-03-07213-7

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

Published electronically:
September 18, 2003

Additional Notes:
Research partially supported by an NSF grant

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2003
American Mathematical Society