Linear operators preserving the numerical radius of matrices

Author:
Chi-Kwong Li

Journal:
Proc. Amer. Math. Soc. **99** (1987), 601-608

MSC:
Primary 15A04; Secondary 15A60

MathSciNet review:
877024

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Abstract: In this note we characterize all the linear operators on the linear space of complex matrices which preserve the numerical radius and all the linear operators on the real linear space of Hermitian matrices which preserve the numerical radius. From our results, we easily deduce V. J. Pellegrini's characterization of all linear operators that preserve the numerical ranges of matrices and the result of Marcus and Moyls concerning the linear operators that preserve the eigenvalues of Hermitian matrices.

**[1]**Chi-Kwong Li, Tin Yau Tam, and Nam-Kiu Tsing,*The generalized spectral radius, numerical radius and spectral norm*, Linear and Multilinear Algebra**16**(1984), no. 1-4, 215–237. MR**769010**, 10.1080/03081088408817624**[2]**Marvin Marcus,*All linear operators leaving the unitary group invariant*, Duke Math. J.**26**(1959), 155–163. MR**0101241****[3]**Marvin Marcus and B. N. Moyls,*Linear transformations on algebras of matrices*, Canad. J. Math.**11**(1959), 61–66. MR**0099996****[4]**F. D. Murnaghan,*On the field of values of a square matrix*, Proc. Nat. Acad. Sci. U.S.A.**18**(1932), 246-248.**[5]**V. J. Pellegrini,*Numerical range preserving operators on a Banach algebra*, Studia Math.**54**(1975), no. 2, 143–147. MR**0388104**

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DOI:
https://doi.org/10.1090/S0002-9939-1987-0877024-7

Article copyright:
© Copyright 1987
American Mathematical Society