An extension of a convexity theorem of the generalized numerical range associated with
Author:
TinYau Tam
Journal:
Proc. Amer. Math. Soc. 127 (1999), 3544
MSC (1991):
Primary 15A60, 22E15
MathSciNet review:
1473680
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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 AuYeung and Tsing's extension of Westwick's convexity result for .
 1.
Judith
D. Sally, Stretched Gorenstein rings, J. London Math. Soc. (2)
20 (1979), no. 1, 19–26. MR 545198
(80k:14006), http://dx.doi.org/10.1112/jlms/s220.1.19
 2.
Yik
Hoi AuYeung and Yiu
Tung Poon, A remark on the convexity and positive definiteness
concerning Hermitian matrices, Southeast Asian Bull. Math.
3 (1979), no. 2, 85–92. MR 564798
(81c:15026)
 3.
Yik
Hoi AuYeung and NamKiu
Tsing, Some theorems on the generalized numerical ranges,
Linear and Multilinear Algebra 15 (1984), no. 1,
3–11. MR
731673 (85c:15039), http://dx.doi.org/10.1080/03081088408817574
 4.
Yik
Hoi AuYeung and NamKiu
Tsing, An extension of the HausdorffToeplitz
theorem on the numerical range, Proc. Amer.
Math. Soc. 89 (1983), no. 2, 215–218. MR 712625
(85f:15021), http://dx.doi.org/10.1090/S00029939198307126254
 5.
Kong
Ming Chong, An induction theorem for rearrangements, Canad. J.
Math. 28 (1976), no. 1, 154–160. MR 0393390
(52 #14200)
 6.
K. E. Gustafson and D. K. M. Rao, Numerical Range: the field of values of linear operators and matrices, Springer, New York, 1997. CMP 97:03
 7.
Albert
W. Marshall and Ingram
Olkin, Inequalities: theory of majorization and its
applications, Mathematics in Science and Engineering, vol. 143,
Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New
YorkLondon, 1979. MR 552278
(81b:00002)
 8.
R. F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of letters, Proc. Edinburgh Math. Soc. 21 (1903), 144157.
 9.
L.
Mirsky, On a convex set of matrices, Arch. Math.
10 (1959), 88–92. MR 0106913
(21 #5643)
 10.
Yiu
Tung Poon, Another proof of a result of Westwick, Linear and
Multilinear Algebra 9 (1980), no. 1, 35–37. MR 586669
(81h:15015), http://dx.doi.org/10.1080/03081088008817347
 11.
Yiu
Tung Poon, Generalized numerical ranges, joint
positive definiteness and multiple eigenvalues, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1625–1634. MR 1372044
(97g:15030), http://dx.doi.org/10.1090/S0002993997037817
 12.
T.Y. Tam, Kostant's convexity theorem and the compact classical groups, Linear and Multilinear Algebra 43 (1997), 87113.
 13.
T.Y. Tam, Generalized numerical ranges, numerical radii, and Lie groups, manuscript, 1996.
 14.
T.Y. Tam, Plotting the generalized numerical range associated with the compact connected Lie groups, manuscript, 1996.
 15.
R.
Westwick, A theorem on numerical range, Linear and Multilinear
Algebra 2 (1975), 311–315. MR 0374936
(51 #11132)
 1.
 M. F. Atiyah and R. Bott, The YangMills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523615. MR 80k:14006
 2.
 Y. H. AuYeung and Y. T. Poon, A remark on the convexity and positive definitness concerning Hermitian matrices, Southeast Asian Bull. Math. 3 (1979), 8592. MR 81c:15026
 3.
 Y. H. AuYeung and N. K. Tsing, Some theorems on the generalized numerical ranges, Linear and Multilinear Algebra, 15 (1984), 311. MR 85c:15039
 4.
 Y. K. AuYeung and N. K. Tsing, An extension of the HausdorffToeplitz theorem on the numerical range, Proc. Amer. Math. Soc. 89, (1983), 215218. MR 85f:15021
 5.
 K. M. Chong, An induction theorem for rearrangements, Canad. J. Math. 28 (1976), 154160. MR 52:14200
 6.
 K. E. Gustafson and D. K. M. Rao, Numerical Range: the field of values of linear operators and matrices, Springer, New York, 1997. CMP 97:03
 7.
 A. W. Marshall and I. Olkin, Inequalities: Theory of majorization and its applications, New York: Academic Press, 1979. MR 81b:00002
 8.
 R. F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of letters, Proc. Edinburgh Math. Soc. 21 (1903), 144157.
 9.
 L. Mirsky, On a convex set of matrices, Arch. Math. 10 (1959), 8892. MR 21:5643
 10.
 Y. T. Poon, Another proof of a result of Westwick, Linear and Multilinear Algebra, 9 (1980), 3537. MR 81h:15015
 11.
 Y. T. Poon, Generalized numerical ranges, joint positive definiteness and multiple eigenvalues, Proc. Amer. Math. Soc. 125 (1997), 16251634. MR 97g:15030
 12.
 T.Y. Tam, Kostant's convexity theorem and the compact classical groups, Linear and Multilinear Algebra 43 (1997), 87113.
 13.
 T.Y. Tam, Generalized numerical ranges, numerical radii, and Lie groups, manuscript, 1996.
 14.
 T.Y. Tam, Plotting the generalized numerical range associated with the compact connected Lie groups, manuscript, 1996.
 15.
 R. Westwick, A theorem on numerical range, Linear and Multilinear Algebra 2 (1975), 311315. MR 51:11132
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Additional Information
TinYau Tam
Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 368495310
Email:
tamtiny@mail.auburn.edu
DOI:
http://dx.doi.org/10.1090/S0002993999046468
PII:
S 00029939(99)046468
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
