论文标题
$ \ MATHBB K $ - 有限对称域上操作员的均匀元组
$\mathbb K$-homogeneous tuple of operators on bounded symmetric domains
论文作者
论文摘要
令$ω$为$ \ Mathbb c^d。$ in Cark $ r $的不可约定的对称域，让$ \ Mathbb k $是Biholomorphic Automormorphic Automormorphic Automormorphic Automormorphic Automormorphic Automormorphic Automormormorphic Automormormorphic Automormormorphic Automormormorphic Automormormorphism of tomain $ω$的标识组件$ G $的最大紧凑型亚组。由线性转换组成的组$ \ mathbb k $自然作用于通勤有限的线性运算符的任何$ d $ -tuple $ \ boldsymbol t =（t_1，\ ldots，t_d）$。如果此操作的轨道模式统一等效是单身人士，那么我们说$ \ boldsymbol t $是$ \ mathbb {k} $ - 同质性。在本文中，我们获得了所有$ \ mathbb {k} $ - 同质$ d $ -tuple $ \ boldsymbol {t} $作为乘务的运营商$ z_1，z_1，z_d $，z_d $ z_d $ od od holomorphic punction $ω$之间。使用此模型，我们获得了（i）有限性的标准，（ii）Cowen-Douglas类（III）单一等价和相似性的成员资格。特别是，我们表明，加权伯格曼空间上乘以坐标函数的$ d $ tuple的伴随在Cowen-Douglas类$ b_1（ω）$中。对于一个有界的对称域$ 2 $的$ω$，给出了操作员$ \ sum_ {i = 1}^d t_i^*t_i $的明确描述。通常，基于此公式，我们做出了一个猜想，给出了该操作员的形式。
Let $Ω$ be an irreducible bounded symmetric domain of rank $r$ in $\mathbb C^d.$ Let $\mathbb K$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $Ω$. The group $\mathbb K$ consisting of linear transformations acts naturally on any $d$-tuple $\boldsymbol T=(T_1,\ldots, T_d)$ of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\boldsymbol T$ is $\mathbb{K}$-homogeneous. In this paper, we obtain a model for all $\mathbb{K}$-homogeneous $d$-tuple $\boldsymbol{T}$ as the operators of multiplication by the coordinate functions $z_1,\ldots ,z_d$ on a reproducing kernel Hilbert space of holomorphic functions defined on $Ω$. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class (iii) unitary equivalence and similarity of these $d$-tuples. In particular, we show that the adjoint of the $d$-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class $B_1(Ω)$. For a bounded symmetric domain $Ω$ of rank $2$, an explicit description of the operator $\sum_{i=1}^d T_i^*T_i$ is given. In general, based on this formula, we make a conjecture giving the form of this operator.
