This is my question: Is the following statement true ?
Let $H$ be a real or complex Hilbertspace and $R,S:H \to H$ compact operators. For every $n\in\mathbb{N}$ the following inequality holds:
$\sum_{j=1}^n s_j(RS) \leq \sum_{j=1}^n s_j(R)s_j(S)$
Note: $s_j(R)$ denotes the j-th singular value of the opeartor $R$. The sequence of the singular values falls monotonically to zero.
With best regards, mat
Edit: I found out, that the statement is true for products instead of sums. By that I mean:
Let $H$ be a $\mathbb{K}$-Hilbertspace and $R,S: H \to H$ compact operators. For every $n\in\mathbb{N}$ we have:
$\Pi_{j=1}^n s_j(RS) \leq \Pi_{j=1}^n s_j(R)s_j(S)$
Is it possible to derive the statement for sums from this?