2-等距的Cauchy对偶次正规性问题的一个解
A solution to the Cauchy dual subnormality problem for 2-isometries
摘要：
柯西对偶次正规性问题是问一个$T$的$2$ -等距的柯西对偶算子$T^{\prime}:=T(T^*T)^{-1}$是否是次正规的。在本文中，我们证明了该问题有一个负解。第一个反例很大程度上依赖于一个重构定理，该定理指出，如果$T$是一个在一个根有向树上的$2$ -等距加权移位，且其非零权重满足扰动核条件，则$T^{\prime}$是次正规的当且仅当$T$满足（未扰动）核条件。第二个反例源于局部有限根有向树的$2$ -等距邻接算子，通过对Cauchy对偶次正规性问题正解的深入研究，得到了该反例.证明了若$T$是满足核条件的$2$等距或拟布朗等距，则$T^{\prime}$是次正规的.构造了一个有根有向树的$2$ -等距邻接算子$T$，使得$T$不满足核条件，$T$不是拟布朗等距算子，且$T^{\prime}$是次正规算子.
英文摘要：
The Cauchy dual subnormality problem asks whether the Cauchy dual operator $T^{\prime}:=T(T^*T)^{-1}$ of a $2$-isometry $T$ is subnormal. In the present paper we show that the problem has a negative solution. The first counterexample depends heavily on a reconstruction theorem stating that if $T$ is a $2$-isometric weighted shift on a rooted directed tree with nonzero weights that satisfies the perturbed kernel condition, then $T^{\prime}$ is subnormal if and only if $T$ satisfies the (unperturbed) kernel condition. The second counterexample arises from a $2$-isometric adjacency operator of a locally finite rooted directed tree again by thorough investigations of positive solutions of the Cauchy dual subnormality problem in this context. We prove that if $T$ is a $2$-isometry satisfying the kernel condition or a quasi-Brownian isometry, then $T^{\prime}$ is subnormal. We construct a $2$-isometric adjacency operator $T$ of a rooted directed tree such that $T$ does not satisfy the kernel condition, $T$ is not a quasi-Brownian isometry and $T^{\prime}$ is subnormal.
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