论文标题
在IV型和矩图的cartan域上通勤托管运营商
Commuting Toeplitz operators on Cartan domains of type IV and moment maps
论文作者
论文摘要
让我们考虑一下，对于$ n \ geq 3 $，cartan域$ \ mathrm {d} _n^{\ mathrm {iv}} $ IV类型IV。在加权的伯格曼空间上，$ \ mathcal {a}^2_λ（\ mathrm {d} _n^{\ mathrm {iv}}）$我们研究了由Toeplitz Operators具有特殊符号的toeplitz Operators生成的交换$ C^*$的问题。我们专注于$ \ Mathrm {d} _nn^{\ Mathrm {\ Mathrm {iv}} $ biholomormormormormormormormormorphisms的子组$ \ Mathrm {so}（so}（n）\ times \ times \ times \ mathrm {so}（2）$。 $ \ mathrm {so}（n）\ times \ times \ mathrm {so}（2）$ - 不变符号产生产生交换性$ c^*$ - 代数的toeplitz操作员，但是当我们考虑在最大的torus或$ \ mathrm下，我们考虑符号是不变的符号时会丢失通勤性。我们计算$ \ mathrm {so}（so}（2）$ - 在$ \ mathrm {d} _n _n^{\ mathrm {iv}} $上被认为是Bergman emplectic歧管的$ \ \ m mathrm {so}（2）$ - 在$ \ mathrm {d}上进行$ \ mathrm {so}（2）$ - compation of $ \ mathrm {so}（2）$ - compation of the Mathrm {so}（2）} $的矩我们证明了表格$ a = f \circμ^{\ mathrm {so}（2）} $的符号空间，由$ l^\ infty（\ m atrm {d} _n^} _n^{\ mathrm {\ mathrm {iv}}}}}^{μ^{μ^{这会产生交换性$ C^*$ - 代数。还获得了这些Toeplitz运算符的光谱积分公式。
Let us consider, for $n \geq 3$, the Cartan domain $\mathrm{D}_n^{\mathrm{IV}}$ of type IV. On the weighted Bergman spaces $\mathcal{A}^2_λ(\mathrm{D}_n^{\mathrm{IV}})$ we study the problem of the existence of commutative $C^*$-algebras generated by Toeplitz operators with special symbols. We focus on the subgroup $\mathrm{SO}(n) \times \mathrm{SO}(2)$ of biholomorphisms of $\mathrm{D}_n^{\mathrm{IV}}$ that fix the origin. The $\mathrm{SO}(n) \times \mathrm{SO}(2)$-invariant symbols yield Toeplitz operators that generate commutative $C^*$-algebras, but commutativity is lost when we consider symbols invariant under a maximal torus or under $\mathrm{SO}(2)$. We compute the moment map $μ^{\mathrm{SO}(2)}$ for the $\mathrm{SO}(2)$-action on $\mathrm{D}_n^{\mathrm{IV}}$ considered as a symplectic manifold for the Bergman metric. We prove that the space of symbols of the form $a = f \circ μ^{\mathrm{SO}(2)}$, denoted by $L^\infty(\mathrm{D}_n^{\mathrm{IV}})^{μ^{\mathrm{SO}(2)}}$, yield Toeplitz operators that generate commutative $C^*$-algebras. Spectral integral formulas for these Toeplitz operators are also obtained.
