An extension of a convexity theorem

of the generalized numerical range

associated with

Author:
Tin-Yau Tam

Journal:
Proc. Amer. Math. Soc. **127** (1999), 35-44

MSC (1991):
Primary 15A60, 22E15

MathSciNet review:
1473680

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Abstract | References | Similar Articles | Additional Information

Abstract: For any , let be the following subset of :

We show that if , then is always convex. When , it is an ellipsoid, probably degenerate. The convexity result is best possible in the sense that if we have defined similarly, then there are examples which fail to be convex when and .

The set is also symmetric about the origin for all , and contains the origin when . Equivalent statements of this result are given. The convexity result for is similar to Au-Yeung and Tsing's extension of Westwick's convexity result for .

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Additional Information

**Tin-Yau Tam**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310

Email:
tamtiny@mail.auburn.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-04646-8

Keywords:
Numerical range,
convexity,
special orthogonal group,
weak majorization

Received by editor(s):
November 26, 1996

Received by editor(s) in revised form:
May 9, 1997

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1999
American Mathematical Society