Factoring Non-negative Operator Valued Trigonometric Polynomials in Two Variables
Factoring Non-negative Operator Valued Trigonometric Polynomials in Two Variables
复制标题
将非负算子值三角多项式因式分解为两个变量
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Michael A. Dritschel
中科院分区：
文献类型：
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作者：
Michael A. Dritschel
利用舒尔补技巧，证明了一个次数为（d1，d2）的二元非负算子值三角多项式可以写成次数至多为（d1，2d2 − 1）的至多2d2个解析多项式的埃尔米特平方的有限和。类似于真实的代数中的塔斯基转移原理，当系数空间是有限维时，证明将问题提升到超积，在那里解决它，然后表明这意味着在原始上下文中存在解。一般结果是通过紧性论证得到的。虽然证明是非建设性的，但它仍然导致一个具体的算法的因式分解。
Using Schur complement techniques, it is shown that a non-negative operator valued trigonometric polynomial in two variables with degree (d1, d2) can be written as a finite sum of hermitian squares of at most 2d2 analytic polynomials with degrees at most (d1, 2d2 − 1). In analogy with the Tarski transfer principle in real algebra, when the coefficient space is finite dimensional, the proof lifts the problem to an ultraproduct, solves it there, and then shows that this implies the existence of a solution in the original context. The general result is obtained through a compactness argument. While the proof is non-constructive, it nevertheless leads to a concrete algorithm for the factorization.