I have been reading the paper - "Introduction to Quantum Fisher Information". In section 1.2 the author talks about the linear map $\mathbb{J}_D$, which he defines as follows:

Let $D \in M_n$ be a positive invertible matrix. The linear mapping $\mathbb{J}^f_D:M_n \to M_n$ is defined by the formula

$\mathbb{J}_D^f=f(\mathbb{L}_D\mathbb{R}^{-1}_D)\mathbb{R}_D$

where $f:\mathbb{R}^+\to\mathbb{R}^+$,

$\mathbb{L}_D(X)=DX$ and $\mathbb{R}_D(X)=XD$.

Also, the author points out that the inverse mapping of $\mathbb{J}_D^f$ is the given by

$\frac{1}{f}(\mathbb{L}_D\mathbb{R}^{-1}_D)\mathbb{R}_D^{-1}$

Then he gives the following example, of which I am able to get the first part but not the inverse map.

*Example 1*: If $f(x)=(x+1)/2$, then

$\mathbb{J}_DB=\frac{1}{2}(DB+BD)$.

$\mathbb{J}_DB=\int_0^\infty \text{exp}(-tD/2)B\text{exp}(-tD/2)dt$

Can you tell me how does he get the required formula for the inverse map? Also, I would like to know what is the meaning of the map $\mathbb{J}_D$? What does the integral of the linear map represent? Can you also provide me some reference book where I could look for these topics?