论文标题
I型cartan域上的径向样toeplitz操作员
Radial-like Toeplitz operators on Cartan domains of type I
论文作者
论文摘要
令$ \ mathrm {d}^\ mathrm {i} _ {n \ times n} $是I型的cartan域，由复杂的$ n \ times n $矩阵$ z $组成，可以满足$ z^*z^z <i_n $。对于一个符号$ a \ in l^\ infty（\ mathrm {d}^\ mathrm {i} _ {n \ times n}）$，我们考虑三个类似径向的类型条件：1）左（右）$ \ mathrm {u}（u}（u}（u}（n））$ - 可以通过$ ncords $ a a（z）定义的n（z）= a \ big（（（z^*z）^\ frac {1} {2} \ big）$（$ a（z）= a \ big（（（zz^*）^\ frac {1} {2} {2} \ big）$，分别为$），和2），2）$ \ mathrm {u}（n）由条件$ a（a^{ - 1} zb）= a（z）$定义，每个$ a，b \ in \ mathrm {u}（n）$。我们证明，对于$ n \ geq 2 $，这些产生了不同的符号集。如果$ a $满足1），则左或向右，$ b $满足2），那么我们证明，相应的Toeplitz运营商$ T_A $和$ T_B $通勤在每个加权伯格曼空间上。此外，在满足的条件1）中，左或右，对于$ n \ geq 2 $，符号$ a $，其相应的toeplitz运算符$ t_a $是非正常的。我们使用这些事实来证明$ n \ geq 2 $的存在，是由Toeplitz Operators生成的可交换banach non-$ c^*$代数。
Let $\mathrm{D}^\mathrm{I}_{n \times n}$ be the Cartan domain of type I which consists of the complex $n \times n$ matrices $Z$ that satisfy $Z^*Z < I_n$. For a symbol $a \in L^\infty(\mathrm{D}^\mathrm{I}_{n \times n})$ we consider three radial-like type conditions: 1) left (right) $\mathrm{U}(n)$-invariant symbols, which can be defined by the condition $a(Z) = a\big((Z^*Z)^\frac{1}{2}\big)$ ($a(Z) = a\big((ZZ^*)^\frac{1}{2}\big)$, respectively), and 2) $\mathrm{U}(n) \times \mathrm{U}(n)$-invariant symbols, which are defined by the condition $a(A^{-1}ZB) = a(Z)$ for every $A, B \in \mathrm{U}(n)$. We prove that, for $n \geq 2$, these yield different sets of symbols. If $a$ satisfies 1), either left or right, and $b$ satisfies 2), then we prove that the corresponding Toeplitz operators $T_a$ and $T_b$ commute on every weighted Bergman space. Furthermore, among those satisfying condition 1), either left or right, there exist, for $n \geq 2$, symbols $a$ whose corresponding Toeplitz operators $T_a$ are non-normal. We use these facts to prove the existence, for $n \geq 2$, of commutative Banach non-$C^*$ algebras generated by Toeplitz operators.
