Sums of Hermitian Squares as an Approach to the BMV Conjecture

Details Sums of Hermitian Squares as an Approach to the BMV Conjecture Sums of Hermitian Squares as an Approach to the BMV Conjecture

2008-02-08

arxiv.org/abs/0802.1153v2

Mathematics

Functional Analysis

9 pages, grammatical corrections, typos added, new references

Scientific paper

Lieb and Seiringer stated in their reformulation of the Bessis-Moussa-Villani (BMV) conjecture that all coefficients of the polynomial p(t)=Tr[(A+tB)^m], where A and B are positive semidefinite matrices of the same size and m an arbitrary integer, are nonnegative. The coefficient of t^k is the trace of S_{m,k}(A,B), which is the sum of all words of length m in the letters A and B in which B appears exactly k times. We consider the case k=4 and show that S_{m,4}(A,B) is a sum of hermitian squares and commutators. In particular, the trace of S_{m,4}(A,B) is nonnegative.

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