$\def\1{\mathbb{1}}$
Suppose we're in a "good enough" (finite for example) space, and we have positive (semi)-definite operators $P$ and $Q$.
Let $\log{(P)}$ and $\log{(P)}$ be logarithms of $P$ and $Q$, defined either as the power series, the logarithm of the Jordan decomposition with the change of basis or the inverse of the exponential function. I'm not sure if these are all equivalent, but let's suppose they are, for now at least.
Then we have the following integral representation
$$ \log{(P)} - \log{(Q)} = \int_0^\infty \Big(\frac{1}{Q + x\1} - \frac{1}{P + x\1}\Big) dx $$
where $\1$ is the identity, and $\frac{1}{P + x\1} = (P+x\1)^{-1}$.
Is there a simple proof of this property? I wouldn't know where to start since I'm not sure about the correct/universal definition of the logarithm of an operator.
Any help is appreciated !
Also I'd be thankful if anyone can direct me to a universal definition of $e^P$ and $\log{P}$. I believe that for $e^P$ the power series is enough and the equivalence to Jordan decomposition + $\operatorname{exp}$ is straightforward, but then for $\log{(P)}$ it doesn't always converge, so maybe there's a different way to define it.
We should assume that $P,Q>0$ in order that $\log P$ and $\log Q$ are well-defined. Suppose we are in a finite dimensional inner product space. Then by the spectral decomposition theorem, there exist unitary matrices $U,V$ such that $$ P=UAU^*,\quad Q=VBV^* $$for some diagonal matrices $A,B>0$. Note that $\log P= U\left(\log A\right) U^*$ and $\log Q =V\left(\log B\right) V^*$. Now, we have $$\begin{eqnarray} \int_0^N (P+x1)^{-1}dx = U\left(\int_0^N (A+x1)^{-1}dx\right)U^*&=&U\left(\log (A+N1)-\log A\right)U^*\\&=&U\left(\log \left(1+\frac{A}{N}\right) \right)U^*-\log P+\log N\cdot 1 \end{eqnarray}$$ and similarly $$ \int_0^N (Q+x1)^{-1}dx=V\left(\log \left(1+\frac{B}{N}\right)\right)V^*-\log Q+\log N\cdot 1. $$ Here, $1$ denotes the identity operator. Hence, $$\begin{eqnarray} \int_0^N (P+x1)^{-1}-(Q+x1)^{-1}dx&=&-\log P+\log Q+\\&&+U\left(\log \left(1+\frac{A}{N}\right)\right)U^*-V\left(\log \left(1+\frac{B}{N}\right)\right)V^*\\ &\to&-\log P+\log Q \end{eqnarray}$$ as $N\to\infty$. This proves $$ \int_0^\infty (P+x1)^{-1}-(Q+x1)^{-1}dx=-\log P+\log Q. $$ If we are in a Hilbert space, then similar proof is possible by the spectral representation $$ P=\int_{(0,\infty)}\lambda dE_1(\lambda), \quad Q=\int_{(0,\infty)}\lambda dE_2(\lambda). $$