ON THE MATRIX CAUCHY-SCHWARZ INEQUALITY

The main goal of this work is to present new matrix inequalities of Cauchy-Schwarz type. In particular, we investigate the so-called Lieb functions, whose definition came as an umbrella of Cauchy-Schwarz-like inequalities, then we consider the mixed Cauchy-Schwarz in-equality. This latter inequality has been influential in obtaining several other matrix inequalities, including numerical radius and norm results. Among many other results, we show that ⠃T ⠃ s 41 (⠃|T|+|T*|+2RT ⠃+⠃|T|+|T*|-2RT ⠃), where RT is the real part of the matrix T .