发表机构
Politecnico di Milano; Graz University of Technology（米兰理工大学; 格拉茨工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对双扇形 Clifford 算子引入谱投影，给出其有界性的两个判据，并应用于广义梯度算子，在常数系数情形下导出显式表示，且将梯度符号与 Clifford-Hilbert 变换相联系。
AI 中文摘要
我们考虑右 Hilbert 模 V 上关于 Clifford 代数 R_n 的右线性算子 T，其 S-谱位于锐双扇形中。对于这些双扇形算子，我们引入与双扇形的两个锥体相关联的谱投影 P_±。它们将 Hilbert 模分解为两个子模 V=V_++V_-，并将双扇形算子 T 分解为两个扇形算子 T|_±。该理论中一个关键但非平凡的基石是投影 P_± 的有界性，而这又与算子 T 的有界 H^∞-函数演算相关。我们提供了两个实用判据：要么平方算子具有有界 H^∞-函数演算，要么该算子是 m-增生算子。最后，我们将这些结果应用于具有非常数系数的梯度算子 ∇_a。对于具有常数系数的特定梯度，我们甚至能够在 Fourier 空间中导出子模 V_± 和投影 P_± 的显式表示。此外，我们将梯度算子的符号与 Clifford-Hilbert 变换等同起来。该符号在向量算子的分数幂中起着核心作用，例如用于热传播的非局部 Fourier 定律。
英文摘要
We consider right-linear operators $T$ on a right Hilbert module $V$ over the Clifford algebra R_n, whose S-spectrum lies in an acute double sector. For these bisectorial operators, we introduce spectral projectors P_\pm associated with the two cones of the double sector. They decompose the Hilbert module into two submodules V=V_++V_-, and the bisectorial operator T into two sectorial operators T|_\pm. A crucial but non-trivial, cornerstone in this theory is the boundedness of the projectors P_\pm, which is, in turn connected to a bounded H^\infty-functional calculus of the operator T. We provide two practical criteria: either the squared operator admits a bounded H^\infty-functional calculus, or the operator is m-accretive. Finally, we apply these results to the gradient operator \nabla_a with nonconstant coefficients. For the particular gradient with constant coefficients, we are even able to derive explicit representations of the submodules V_\pm and the projectors P_\pm in Fourier space. Moreover, we identify the sign of the gradient operator with the Clifford-Hilbert transform. This sign plays a central role in the fractional powers of vector operators, which are used, for instance, in the non-local Fourier law of heat propagation.
