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I have asked this question on MathSE and someone advised me to ask it here. The link is .

I'm studying the paper Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators by Changsoo Bahn and Chul Ki Ko and Yong Moon Park (arxiv link).

I have a question on the proof (p.15) of theorem 3.1 (p.10) of this paper, in which a part of deduction goes as

\begin{align} &[W^2-\sum_{l=1}^d(W_l)_l,W_k\partial_k+\partial_kW_k]\\=&[W^2,W_k\partial_k+\partial_kW_k]-[\sum_{l=1}^d(W_l)_l,W_k\partial_k+\partial_kW_k]\\=&-2W_k((W^2)_k-\sum_{l=1}^d(W_l)_{lk}) \end{align}

where $W=(W_1,\cdots,W_d)$ is a function $:\mathbb{R}^d \to \mathbb{R}^d$, $(W_i)_k$ is the $k^{th}$ derivative of $W_i$ and $W^2=\sum_{i=1}^d (W_i)^2$.

I am stuck in the last equal sign, for which I think we need for $A,B \in \mathcal{D}$:

$$[A,B\partial_k+\partial_k B]=-2B(A)_k.$$

But my calculation for this goes as:

\begin{align} &[A,B\partial_k+\partial_k B]+2B(A)_k\\=&AB\partial_k+A\partial_kB-B\partial_kA-\partial_kBA+2B\partial_kA-2BA\partial_k\\\not = &0 \end{align}

Am I missing something or is the paper wrong?

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  • $\begingroup$ What are $W,W_{k}$?Is $W_{k}$ the partial derivative (in direction $k$) of $W$? $\endgroup$
    – Asaf
    Commented Aug 31, 2018 at 17:37
  • $\begingroup$ @Asaf sorry about the confusion. I've made some changes to the description of the question $\endgroup$
    – aeei.w.1995
    Commented Aug 31, 2018 at 19:32

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