An estimate on the trace norm of a product of operators

Question

I am having some troubles understanding the following paper of Demuth, Stollmann, Stolz and Van Casteren that improves the Hölder inequality for the trace norm : https://link.springer.com/article/10.1007/BF01197534

Theorem 1 in this paper gives an estimate of the trace norm of a product of two kernel operators $ \boldsymbol{A}, \boldsymbol{B} $ with respective kernels $ A, B $ acting on the space $ L^2 \equiv L^2(X, \mathfrak{U}, m) $. The theorem states that if $ A(x, \cdot) $ and $ B(\cdot, x) \in L^2 $ for a.e. $x$, and if $ \int_X \vert\!\vert A(x, \cdot)\vert\!\vert_{L^2} \vert\!\vert B(\cdot, x)\vert\!\vert_{L^2} dm(x) < \infty $, then, there exists a trace class operator $ \boldsymbol{A B} $ with kernel given by the usual composition $ (x, y) \mapsto \int_X A(x, t) B(t, y) dm(t) $ such that

$$ \vert\!\vert \boldsymbol{A B} \vert\!\vert_{tr} \leq \int_X \vert\!\vert A(x, \cdot)\vert\!\vert_{L^2} \vert\!\vert B(\cdot, x)\vert\!\vert_{L^2} dm(x) $$

The proof then proceeds to show that the product $ \boldsymbol{A B} $ can be turned into a product of two Hilbert-Schmidt operators (which is fine for me), and does not prove the estimate, which certainly implies that it is trivial. Nevertheless, I don't really know how to start. The several definitions and characterisations of the trace norm $ \vert\!\vert \boldsymbol{A } \vert\!\vert_{tr} := \mathrm{tr}( \sqrt{ \boldsymbol{A}^* \boldsymbol{A} } ) $ do not show any integral formula for such a norm (unless I missed this point). The trace is of course given by $ \mathrm{tr}( \boldsymbol{A}) = \int_X A(x, x) dm(x) $, but this does not imply anything on the trace norm, as far as I know.

Any help is welcome to understand how they get their estimate.

Answer 1

From the polar decomposition, there exists a partial isometry $V$ such that $|AB|=VAB$. Then, using Cauchy-Schwarz and the fact that $$ VA\xi=V\,\int_X\xi(x)\,A(\cdot,x)\,dm(x)=\int_X\xi(x)\,VA(\cdot,x)\,dm(x) $$ due to the continuity of $V$, \begin{align} \|AB\|_{\rm tr} &=\operatorname{Tr}(VAB) =\int_X(VAB)(x,x)\,dm(x)\\[0.3cm] &=\int_X\int_X(VA)(x,y)B(y,x)\,dm(y)\,dm(x)\\[0.3cm] &\leq\int_X\|VA(x,\cdot)\|_2\,\|B(\cdot,x)\|_2\,dm(x)\\[0.3cm] &\leq\int_X\|A(x,\cdot)\|_2\,\|B(\cdot,x)\|_2\,dm(x).\\[0.3cm] \end{align}