一类齐次算子的曲率计算
Curvature calculations for a class of homogeneous operators
摘要：
对于在\cite{CD}中引入的类${\mathrm B}_n(Ω)$中的算子$T$，相应丛$E_T$的曲率共变导数直到某个阶的同时酉等价类确定了算子$T$的酉等价类。在论文\cite{CD2}中，作者提出了一个问题：是否存在一对不等价的算子$T_1$和$T_2$，使得除最高阶导数外，曲率及其所有共变导数的同时酉等价类一致.在这里，我们表明，一些共变导数是必要的，以确定酉等价类的运营商在${\mathrm B}_n(Ω)$。我们的例子由齐次算子组成。对于齐次算子，曲率及其所有共变导数的同时酉等价类由它们在0处的同时酉等价类确定。这表明计算所有不变量并在一个点（比如0）比较它们就足够了。这些计算，然后进行了一些例子。
英文摘要：
For an operator $T$ in the class ${\mathrm B}_n(Ω)$, introduced in \cite{CD}, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle $E_T$ determine the unitary equivalence class of the operator $T$. In the paper \cite{CD2}, the authors ask if there exists some pair of inequivalent oprators $T_1$ and $T_2$ for which the simultaneous unitary equivalence class of the curvature along with all the covariant derivatives coincide except for the derivative of the highest order. Here we show that some of the covariant derivatives are necessary to determine the unitary equivalence class of the operators in ${\mathrm B}_n(Ω)$. Our examples consist of homogeneous operators. For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives are determined from the simultaneous unitary equivalence class of these at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
