Trace-positive polynomials and the quartic tracial moment problem
نویسندگان
چکیده
منابع مشابه
The tracial moment problem and trace-optimization of polynomials
The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f , what is the smallest trace f(A) can attain for a tuple of matrices A? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives effectively computable bou...
متن کاملTrace-positive polynomials, sums of hermitian squares and the tracial moment problem
A polynomial f in non-commuting variables is trace-positive if the trace of f(A) is positive for all tuples A of symmetric matrices of the same size. The investigation of trace-positive polynomials and of the question of when they can be written as a sum of hermitian squares and commutators of polynomials are motivated by their connection to two famous conjectures: The BMV conjecture from stati...
متن کاملTrace-positive polynomials and the quartic tracial moment problem Polynômes avec une trace positive et le problème quartique des moments traciaux
The tracial analog of Hilbert’s classical result on positive binary quartics is presented: a trace-positive bivariate noncommutative polynomial of degree at most four is a sum of hermitian squares and commutators. This is applied via duality to investigate the truncated tracial moment problem: a sequence of real numbers indexed by words of degree four in two noncommuting variables with values i...
متن کاملThe Truncated Tracial Moment Problem
We present tracial analogs of the classical results of Curto and Fialkow on moment matrices. A sequence of real numbers indexed by words in non-commuting variables with values invariant under cyclic permutations of the indexes, is called a tracial sequence. We prove that such a sequence can be represented with tracial moments of matrices if its corresponding moment matrix is positive semidefini...
متن کاملPositive Polynomials, Sums of Squares and the Moment Problem
Acknowledgements A lot of people helped me during the work on this thesis. First of all, I am greatly indebted to Alexander Prestel, Claus Scheiderer and Markus Schweighofer. I benefitted a lot from their constant logistic and mathematical support. I also want to thank Robert Denk, David Grimm and Daniel Plaumann for many interesting discussions on the topic of this work. In 2007, I enjoyed a s...
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ژورنال
عنوان ژورنال: Comptes Rendus Mathematique
سال: 2010
ISSN: 1631-073X
DOI: 10.1016/j.crma.2010.06.005