论文标题
具有简单光谱的非紧凑型Hankel操作员的逆频谱问题
An inverse spectral problem for non-compact Hankel operators with simple spectrum
论文作者
论文摘要
我们认为，对于一类非压缩Hankel运算符$ H $的反向频谱问题，以使$ H $的模量（仅限于正交补充的内核）具有简单的光谱。与紧凑型操作员的情况类似，我们证明了独特性结果，即，我们证明了同类产品中的汉克尔运营商是由频谱数据唯一决定的。换句话说，将Hankel运算符映射到频谱数据的光谱图是Injective。此外，与紧凑型情况相反，我们证明了光谱图的溢流性失败，即，我们证明并非某个自然集合的所有光谱数据都与Hankel oberators相对应。我们在描述光谱图的图像方面取得了一些进展。我们还将应用于CubicSzegő方程式。特别是，我们证明并非所有具有BMOA初始数据的解决方案几乎是周期性的。这与VMOA初始数据的已知结果形成鲜明对比。
We consider an inverse spectral problem for a class of non-compact Hankel operators $H$ such that the modulus of $H$ (restricted onto the orthogonal complement to its kernel) has simple spectrum. Similarly to the case of compact operators, we prove a uniqueness result, i.e. we prove that a Hankel operator from our class is uniquely determined by the spectral data. In other words, the spectral map, which maps a Hankel operator to the spectral data, is injective. Further, in contrast to the compact case, we prove the failure of surjectivity of the spectral map, i.e. we prove that not all spectral data from a certain natural set correspond to Hankel operators. We make some progress in describing the image of the spectral map. We also give applications to the cubic Szegő equation. In particular, we prove that not all solutions with initial data in BMOA are almost periodic; this is in a sharp contrast to the known result for initial data in VMOA.
