论文标题
单位球中繁殖的内核希尔伯特空间与有限套装的几何形状之间的距离
Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball
论文作者
论文摘要
在本文中，我们研究了繁殖的内核希尔伯特空间，其乘数代数与它们所处的点集的几何形状之间的关系。我们引入了一个适合测量繁殖核希尔伯特空间之间距离的Banach-Mazur距离的变体，该距离量化了两个空间距离同构同构为复制核Hilbert Space的距离。我们引入了乘数代数的类似距离，该距离量化了两个代数距离完全异构形构的多远。我们表明，在Drury-Arveson空间的有限维数商的设置中，两个空间在且仅当它们的乘数代数是“接近”时彼此“接近”，并且只有当且仅当下面的点集是“几乎一致”时，就会发生这种情况，这是一个几乎一致的，这意味着一个集合非常接近另一个单元的图像，这是一个在Biholomorphormorpholomorpholomorphormorphormorphormorphol的图像中，并且这是一个非常接近的图像。这些等效是作为我们证明的定量估计值的推论。
In this paper we study the relationships between a reproducing kernel Hilbert space, its multiplier algebra, and the geometry of the point set on which they live. We introduce a variant of the Banach-Mazur distance suited for measuring the distance between reproducing kernel Hilbert spaces, that quantifies how far two spaces are from being isometrically isomorphic as reproducing kernel Hilbert spaces. We introduce an analogous distance for multiplier algebras, that quantifies how far two algebras are from being completely isometrically isomorphic. We show that, in the setting of finite dimensional quotients of the Drury-Arveson space, two spaces are "close" to one another if and only if their multiplier algebras are "close", and that this happens if and only if the underlying point-sets are "almost congruent", meaning that one of the sets is very close to an image of the other under a biholomorphic automorphism of the unit ball. These equivalences are obtained as corollaries of quantitative estimates that we prove.
