Majorization and Singular Values II

Let $B,D_j ,E_j ,j = 1,2, \cdots ,k,$ be $n \times n$ complex matrices. It is shown that \[ \sigma \left( \sum_{j = 1}^k D_j BE_j^ * \right) \prec_w \sigma (B)\bullet\delta \] where $\delta $ is any vector with components $\delta _1 \geqq \cdots \geqq \delta _n $ that weakly majorizes both the following vectors: \[ \sigma \left( \sum D_j D_j^ * \right)^{1/2}\bullet \sigma \left( \sum E_j E_j^ * \right)^{1/2} \qquad \text{and}\qquad \sigma \left( \sum D_j^ * D_j \right)^{1/2}\bullet \sigma \left( \sum E_j^ * E_j \right)^{1/2} .\] Here $\sigma ( \cdot )$ denotes the vector of singular values arranged in nonincreasing order, $ \prec _w $ denotes weak majorization, and $\bullet$ indicates Schur (entrywise) multiplication. The result unifies several known results concerning majorization statements for singular values.