标量函数的可微性应用于埃尔米特算子--一种傅里叶变换方法
Differentiability of scalar functions applied to Hermitian operators - a Fourier transform approach
摘要：
设g是实数上的一个（即充分可微的）函数。我们知道如何将g应用于C* 代数的埃尔米特元素A。然而，映射A到g（A）的可微性问题并不是微不足道的，因为一般来说“A和dA不交换”。然而，由于该映射线性地依赖于g，因此可以通过傅里叶变换将一般g的情况简化为指数函数的情况。对于后者，有一个显式的n阶导数公式（比标量情况更复杂-仍然是“A和dA不对易”）。通过这种方式，可以通过涉及g在[-r，r]上的（n+1）阶导数的Sobolev范数来限制r球上（A to g（A））的n阶导数的范数。
英文摘要：
Let g be a (say, sufficiently differentiable) function on the reals. One knows how to apply g to Hermitian elements A of a C* algebra. Yet the question of differentiability of the mapping A to g(A) is not trivial, since in general "A and dA do not commute". However, since this mapping depends linearly on g, one can, via Fourier transform, reduce the case of general g to the case of the exponential function. For the latter one has an explicit formula for the n-th derivative (more complicated than in the scalar case - still "A and dA do not commute"). In this way one bounds the norm of the n-th derivative of (A to g(A)) on the r-ball by a Sobolev norm involving the (n+1)-th derivative of g on [-r,r].
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