A LEWENT TYPE DETERMINANTAL INEQUALITY

Abstract

We prove a Lewent type determinantal inequality: Let $A_i$, $i=1,\ldots, n$, be (strictly) contractive trace class operators over a separable Hilbert space. Then \[ \left|\det\left(\frac{I+\displaystyle\sum_{i=1}^n\lambda_iA_i}{I-\displaystyle\sum_{i=1}^n\lambda_iA_i}\right)\right|\le\prod_{i=1}^n\det\left(\frac{I+|A_i|}{I-|A_i|}\right)^{\lambda_i}, \] where $\sum_{i=1}^n \lambda_i = 1$, $\lambda_i \ge 0$, $i=1,\ldots, n$, are (scalar) weights and $|A| = (A^*A)^{1/2}$.