论文标题
统一集团下的乘法运算符的元组
Commuting Tuple of Multiplication Operators Homogeneous under the Unitary Group
论文作者
论文摘要
令$ \ mathcal u（d）$为$ d \ times d $统一矩阵的组。我们发现条件以确保$ \ Mathcal u（d）$ - 均质$ d $ -tuple $ \ boldsymbol t $单位等同于通过坐标函数乘以某些再现核hilbert space $ \ mathcal h_k（\ mathbb b_d，\ mathb b_d，\ mathbb c^n） Hol}（\ Mathbb b_d，\ Mathbb C^n）$，$ n = \ dim \ cap_ {j = 1}^d \ ker t^*_ {j {j}。 u（d）$。我们对$ \ Mathcal u（d）$转换的准不变内核进行分类，并使用两种特定的乘数选择。证据的关键要素是，$ su（d）$恰好具有两个不等的尺寸$ d $的不可约合的统一表示，而无需尺寸$ 2，\ ldots，d-1 $，$ d \ d \ geq 3 $。我们获得了这些操作员之间的界限，降低性和相互统一等效性的明确标准。
Let $\mathcal U(d)$ be the group of $d\times d$ unitary matrices. We find conditions to ensure that a $\mathcal U(d)$-homogeneous $d$-tuple $\boldsymbol T$ is unitarily equivalent to multiplication by the coordinate functions on some reproducing kernel Hilbert space $\mathcal H_K(\mathbb B_d, \mathbb C^n) \subseteq \mbox{\rm Hol}(\mathbb B_d, \mathbb C^n)$, $n= \dim \cap_{j=1}^d \ker T^*_{j}.$ We describe this class of $\mathcal U(d)$-homogeneous operators, equivalently, non-negative kernels $K$ quasi-invariant under the action of $\mathcal U(d)$. We classify quasi-invariant kernels $K$ transforming under $\mathcal U(d)$ with two specific choice of multipliers. A crucial ingredient of the proof is that the group $SU(d)$ has exactly two inequivalent irreducible unitary representations of dimension $d$ and none in dimensions $2, \ldots , d-1$, $d\geq 3$. We obtain explicit criterion for boundedness, reducibility and mutual unitary equivalence among these operators.
