Generalized numerical ranges, joint positive definiteness and multiple eigenvalues

Poon, Yiu

doi:10.1090/S0002-9939-97-03781-7

Generalized numerical ranges, joint positive definiteness and multiple eigenvalues
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Proc. Amer. Math. Soc. 125 (1997), 1625-1634 Request permission

Abstract:

We prove a convexity theorem on a generalized numerical range that combines and generalizes the following results: 1) Friedland and Loewy’s result on the existence of a nonzero matrix with multiple first eigenvalue in subspaces of hermitian matrices, 2) Bohnenblust’s result on joint positive definiteness of hermitian matrices, 3) the Toeplitz-Hausdorff Theorem on the convexity of the classical numerical range and its various generalizations by Au-Yeung, Berger, Brickman, Halmos, Poon, Tsing and Westwick.

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