INNER PRODUCT INEQUALITIES THROUGH CARTESIAN DECOMPOSITION WITH APPLICATIONS TO NUMERICAL RADIUS INEQUALITIES.

This paper intends to show several inner product inequalities using the Cartesian decomposition of the operator. We utilize the obtained results to get norm and numerical radius inequalities. Our results extend and improve some earlier inequalities. Among other inequalities, it is revealed that if T is a n × n complex matrix with the imaginary part ℑT = T--T*/2i, then 1/2 max(∥TT* -- iℑT²∥^1/2, ∥T*T + iℑT²∥^1/2) ≤ ω(T) which is a significant improvement of the classical inequality 1/2 ℑTℑ ≤ ω(T). [ABSTRACT FROM AUTHOR]