Given two (possibly rectangular) diagonal matrices $\Sigma_\text{L}$ and $\Sigma_\text{R}$ with nonnegative elements, what can we say about the singular value decomposition of $$\Sigma_\text{L} X \Sigma_\text{R}$$ where $X \in \mathrm{U}(n)$ is an arbitrary unitary matrix?
Edit: Is there any relationship/bound on the singular value? Any hint is very much appreciated.
The only bounds that apply here are those that apply to the singular values of $AB$ (in terms of the singular values of $A$ and the singular values of $B$). Indeed, using the SVD of $A$ and $B$, we can reduce any such product to a case like this. If $A = U_1 \Sigma_1 V_1^T$ and $B = U_2 \Sigma_2 V_2^T$ are singular value decompositions, then $AB$ has the same singular values as the matrix $$ U_1^T(AB)V_2 = \Sigma_1 (V_1^TU_2) \Sigma_2. $$ One such result is as follows. For the specific case that $A$ and $B$ are square of size $n$, if $s_1(A) \geq \cdots \geq s_n(A)$ denote the singular values of $A$, then $$ \prod_{j=1}^k s_j(AB) \leq \prod_{j=1}^k s_j(A)s_j(B), \quad 1 \leq k \leq n, $$ with equality in the case that $k = n$. A more general result (cf. Theorem III.4.5 in Bhatia's Matrix Analysis, attributed to Gelfand and Naimark) is that for any $\leq i_1 < \cdots < i_k \leq n$, we have $$ \prod_{j=1}^k s_j(AB) \leq \prod_{j=1}^k s_j(A)\prod_{j=1}^n s_{i_j}(B), \quad 1 \leq k \leq n. $$