Polynomials and numerical ranges

Author:
Chi-Kwong Li

Journal:
Proc. Amer. Math. Soc. **104** (1988), 369-373

MSC:
Primary 15A60; Secondary 15A69, 26C10

MathSciNet review:
962800

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an complex matrix. For we study the inclusion relation for the following polynomial sets related to the matrix .

(a) The classical numerical range of the th compound of the matrix .

(b) The th decomposable numerical range of the matrix .

(c) The convex hull of the set of all monic polynomials of degree that divide the characteristic polynomial of . Moreover, we give an example showing that the set described in (a) is not convex in general. This settles a question raised by C. Johnson.

**[1]**Yik Hoi Au-Yeung and Nam-Kiu Tsing,*An extension of the Hausdorff-Toeplitz theorem on the numerical range*, Proc. Amer. Math. Soc.**89**(1983), no. 2, 215–218. MR**712625**, 10.1090/S0002-9939-1983-0712625-4**[2]**Ky Fan and Gordon Pall,*Imbedding conditions for Hermitian and normal matrices*, Canad. J. Math.**9**(1957), 298–304. MR**0085216****[3]**Charles R. Johnson,*Interlacing polynomials*, Proc. Amer. Math. Soc.**100**(1987), no. 3, 401–404. MR**891133**, 10.1090/S0002-9939-1987-0891133-8**[4]**M. Marcus,*Finite dimensional multilinear algebra*. I and II, Marcel Dekker, 1973 and 1975.**[5]**Marvin Marcus and Ivan Filippenko,*Linear operators preserving the decomposable numerical range*, Linear and Multilinear Algebra**7**(1979), no. 1, 27–36. MR**523645**, 10.1080/03081087908817256

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0962800-3

Keywords:
Numerical range,
decomposable numerical range,
characteristic polynomial

Article copyright:
© Copyright 1988
American Mathematical Society