Loewner's Theorem on Monotone Matrix Functions
Loewner's Theorem on Monotone Matrix Functions
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DOI：
10.1007/978-3-030-22422-6
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发表时间：
2019-08
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影响因子：
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通讯作者：
B. Simon
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作者：
B. Simon
这本书是Loewner定理的一首情诗。还有其他数学爱情诗，虽然不多。一个迹象，并不总是存在的，也不是万无一失的，是像这本书，作者已经包括了一些主要人物的图片，在发展的主题正在讨论。一个明显的迹象是读者的第一反应是“怎么会有一整本书都是关于这个主题的”（尽管不是所有的窄书都是爱情诗）。Loewner定理涉及单调矩阵函数的理论，即函数f使得A≤ B f（A）≤ f（B）对于自伴矩阵对。这是一个微妙的概念，可以从以下事实看出（见推论14.3）：如果f：R→ R在所有2× 2矩阵对上是单调的，那么f是仿射的！因此Loewner意识到需要固定一个适当的区间（a，B）R，要求f：（a，B）→ R，并且只要求特征值都在（a，B）中的对A和B的单调性结果。1934年，Charles Loewner证明了一个显著的结果：f在（a，B）上是矩阵单调的当且仅当f在（a，B）上是真实的解析的，并且在上半平面上有一个解析延拓，在那里有一个正虚部。具有这种性质的函数是矩阵单调的，这一点可以从赫格洛兹表示定理中得到，所以我称之为简单的一半。另一个方向是硬的一半。许多应用Loewner的概念矩阵单调函数涉及明确的例子，所以只有容易的一半。矩阵单调性是一个代数命题，但Loewner定理说它等价于一个分析事实。这门学科的魅力之一是它的研究中代数和分析的混合。
This book is a love poem to Loewner’s theorem. There are other mathematical love poems, although not many. One sign, not always present and not foolproof, is that like this book, the author has included pictures of some of the main figures in the development of the subject under discussion. The telltale sign is that the reader’s initial reaction is “how can there a whole book on that subject”(although not all narrow books are love poems).Loewner’s theorem concerns the theory of monotone matrix functions, ie functions, f so that A≤ B⇒ f (A)≤ f (B) for pairs of selfadjoint matrices. That this is a subtle notion is seen by the fact (see Corollary 14.3) that if f: R→ R is monotone on all pairs of 2× 2 matrices, then f is affine! So Loewner realized one needed to fix a proper interval (a, b)⊂ R, demand that f:(a, b)→ R, and only demand the monotonicity result for pairs A and B all of whose eigenvalues are in (a, b). In 1934, Charles Loewner proved the remarkable result that f on (a, b) is matrix monotone on all such n× n pairs (for all n) if and only if f is real analytic on (a, b) and has an analytic continuation to the upper half plane with a positive imaginary part there. That functions with this property are matrix monotone follows in a few lines from the Herglotz representation theorem, so I call that half the easy half. The other direction is the hard half. Many applications of Loewner’s notion of matrix monotone functions involve explicit examples and so only the easy half. Matrix monotonicity is an algebraic statement, but Loewner’s theorem says it is equivalent to an analytic fact. One fascination of the subject is the mix of the algebraic and the analytic in its study.