标题： Clifford共轭算子的S-函数演算 显示英文标题

标题： The S-functional calculus for the Clifford adjoint operator

摘要： 在本文中，我们是在 Clifford 代数 $\mathbb{R}_d$上的模结构框架内进行研究的。我们的研究重点是 Clifford 算子 T 的伴随算子 T* 的 S-谱及其各种形式的 S-函数演算。我们提出的关键结果之一是，T 的双射性可以传递到 T*。这是基于这样一个事实，即对于 Clifford 算子来说，伴随算子 T* 的 S-谱与 T 的 S-谱是相同的。此外，我们证明了对于现有的 S-函数演算的表述，包括有界、无界、$\omega$和$H^\infty$形式，T 的左函数演算与 T* 的右函数演算之间存在明确的联系。这种 T 的左函数演算与 T* 的右函数演算之间的显式联系以及反之亦然，是通过函数$f^\#(s):=\overline{f(\overline{s})}$得到的。最后，我们讨论了这样一个事实，即与二阶算子 T^2-2s_0T+|s|^2 的可逆性相关的 S-谱实际上可以通过一阶$\mathbb{R}$-线性算子 T-I^Rs 的可逆性来表征。 显示英文摘要

摘要： In this paper, we work within the framework of modules over the Clifford algebra $\mathbb{R}_d$. Our investigation focuses on the S-spectrum and the S-functional calculus in its various forms for the adjoint T* of a Clifford operator T. One of the key results we present is that the bisectoriality of T can be transferred to T*. This is grounded in the fact that, for Clifford operators, the S-spectrum of the adjoint operator T* is identical to that of T. Furthermore, we demonstrate that for the existing formulations of the S-functional calculus, including bounded, unbounded, $\omega$, and $H^\infty$ versions, there is a clear connection between the left functional calculus of T and the right functional calculus of T*. This explicit link between the left functional calculus of T and the right functional calculus of T* and vice versa is obtained using the function $f^\#(s):=\overline{f(\overline{s})}$. Finally, we discuss the fact that the S-spectrum, related to the invertibility of the second-order operator T^2-2s_0T+|s|^2, can actually be characterized through the invertibility of the first-order $\mathbb{R}$-linear operator T-I^Rs.