Bulletin Boards

Problem of the week

Let \( A, B \) are \( n \times n \) Hermitian matrices and \( p, q \in [1, \infty] \) with \( \frac{1}{p} + \frac{1}{q} = 1 \). Prove that \[ | Tr (AB) | \leq \| A \|_{S^p} \| B \|_{S^q}. \] (Here, \(\| A \|_{S^p} \) is the \(p\)-Schatten norm of \( A \), defined by \[ \| A \|_{S^p} = \left( \sum_{i=1}^n |\lambda_i|^p \right)^{1/p}, \] where \( \lambda_1, \lambda_2, \dots, \lambda_n \) are the eigenvalues of \( A \).)