I was wondering if the following inequality is true:
Let $\xi_1,...,\xi_n$ be vectors in a Hilbert space $H$ and let $x_{i,j}$ be complex numbers such that $\prod x_{i,j}$ is real and $$\prod x_{i,j}\le 1.$$
Then the sum $\sum_{i,j} x_{i,j}\langle \xi_i, \xi_j\rangle$ is a real number and $$\sum_{i,j} x_{i,j}\langle \xi_i, \xi_j\rangle\le \sum_{i,j}\langle \xi_i,\xi_j\rangle.$$
If so, I would be grateful for a proof/reference.
This inequality arose when I was trying to obtain certain estimate for operators on exponential vectors.
As a possible motivation for the question, consider $H=\mathbb{C}$ and let $x_{i,j}$ be the matrix of an operator $X$. Then the inequality I'm trying to prove just says that $X$ is, in the operator order, less or equal to the identity operator.
A counterexample with vectors from $\mathbb{R}^2$: $\xi_1 = (1,0), \xi_2 = (0,1)$ and $x_{1,1} = 2, x_{2,2} = 1, x_{1,2} = x_{2,1} = 1/2$. Your LHS is then $3$ while the RHS is $2$ and $3 \leq 2$ is false.
P.S.: The "Then the sum ... is a real number" part might still be true, I don't know.