Page 125 - Textos de Matemática Vol. 39

PARTIAL SPECTRA OF SUMS OF HERMITIAN MATRICES
JOA˜O FILIPE QUEIRO´
Dedicated to Eduardo Marques de Sa´, with the highest admiration.
Abstract. Some remarks are made concerning the problem of describing the possible partial spectra of a sum of two Hermitian matrices with given eigen- values.
1. Introduction
Throughout this paper, α = (α1,...,αn) and β = (β1,...,βn) denote two n-tuples of real numbers ordered so that α1 ≥ ··· ≥ αn and β1 ≥ ··· ≥ βn.
The question to be addressed (suggested in [4]) is the following. Fix an integer s ∈ {1,...,n}, and indices k1,...,ks such that 1 ≤ k1 < ··· < ks ≤ n. Given an s-tuple γ = (γk1,...,γks), with γk1 ≥ ··· ≥ γks, when do there exist Hermitian A and B, with spectra α and β respectively, such that γ is a part of the spectrum of A + B? In other words, what are the possible s-tuples γ = (γk1 , . . . , γks ) such that, for j = 1, . . . , s, the j-th coordinate of γ, γkj , is the kj-th eigenvalue of a sum A+B, A and B Hermitian with spectra α and β? We will make some elementary remarks on this problem concerning particular values of s.
2. The full spectrum case
The case where s = n, i.e. we are interested in the possible (complete) spectra of sums A + B, A and B Hermitian with the given spectra α and β, has a long history, with connections to different parts of Mathematics, and has been solved a few years ago. We briefly recall this solution. (The interested reader can find more details in the fine surveys [2, 4].)
Denote by E(α,β) the set of possible such spectra γ. This set is easily seen to be compact and connected, as it is the image of the unitary group under a continuous mapping. It is contained in the hyperplane defined by the trace condition, which we abbreviate to Σγ = Σα + Σβ. In [3] it was shown that it is convex, using the convexity properties of the moment mapping from symplectic geometry.
2000 Mathematics Subject Classification. 15A42, 15A57.
Key words and phrases. Eigenvalues, Hermitian matrix, partial spectrum. This work was supported by CMUC/FCT.
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