实Smirnov函数的值域与化合价
The range and valence of a real Smirnov function
摘要：
本文给出了实Smirnov函数（开单位圆上两个有界解析函数的乘积，其中分母是外的，且径向边界值在单位圆上几乎处处为实）的可能值域的完整描述.我们的技术使用无界对称Toeplitz算子理论，一些一般理论的无界对称算子，经典的Hardy空间，和应用的一致化定理。此外，我们完全刻画了这些实Smirnov函数的可能价时，价是有限的。为了做到这一点，我们构造黎曼曲面，我们称之为圆盘树，方法是将单位圆盘及其在黎曼球面中的补数的副本焊接在一起。我们还利用了某些树，我们称之为价树，它反映了磁盘树的结构。
英文摘要：
We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees by welding together copies of the unit disk and its complement in the Riemann sphere. We also make use of certain trees we call valence trees that mirror the structure of disk trees.
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