单调算子函数、间隙和幂矩问题
Monotone operator functions, gaps and power moment problem
摘要：
本文研究了连续维全矩阵代数的矩阵单调函数类$P_{n+1}(I)$和$P_{n}(I)$之间的间隙函数类。在本文中，我们解决的问题，属于间隙$P_{n}(I) \setminus P_{n+1}(I)$有界区间$I$的特征多项式。我们发现，这个问题的解决方案是密切相关的截断矩问题，汉克尔矩阵和汉克尔扩展的解决方案。也就是说，我们表明，使用截断矩问题的解决方案，我们可以构造连续的间隙中的许多多项式。我们还通过几个例子提供了一些初步的见解，进一步的问题，描述多项式的差距，不是来自截断矩问题。此外，在这篇文章中，我们进一步深化在另一种方式到结构的类矩阵单调函数和差距，他们之间的考虑问题的位置在差距的某些有趣的子类矩阵单调函数出现在连接到插值空间和证明的L{<$}维纳定理整表示的算子单调函数。
英文摘要：
The article is devoted to investigation of the classes of functions belonging to the gaps between classes $P_{n+1}(I)$ and $P_{n}(I)$ of matrix monotone functions for full matrix algebras of successive dimensions. In this paper we address the problem of characterizing polynomials belonging to the gaps $P_{n}(I) \setminus P_{n+1}(I)$ for bounded intervals $I$. We show that solution of this problem is closely linked to solution of truncated moment problems, Hankel matrices and Hankel extensions. Namely, we show that using the solutions to truncated moment problems we can construct continuum many polynomials in the gaps. We also provide via several examples some first insights into the further problem of description of polynomials in the gaps that are not coming from the truncated moment problem. Also, in this article, we deepen further in another way into the structure of the classes of matrix monotone functions and of the gaps between them by considering the problem of position in the gaps of certain interesting subclasses of matrix monotone functions that appeared in connection to interpolation of spaces and in a prove of the L{ö}wner theorem on integral representation of operator monotone functions.
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