Cauchy-Schwarz Inequalities in Hilbert Modules

Adriana Popovici, Dan Popovici
Department of Mathematics
University of the West of Timisoara
B-dul V.Pârvan nr. 4
1900 Timisoara, România
danp@math.uvt.ro, apopovici@yahoo.com

The aim in this paper is to study various types of Cauchy-Schwarz inequalities and related topics in Hilbert modules $(E,[\cdot ,\cdot])$ over $\mathbb{C}^*$-algebras $A$. The results are qualitatively improved if, in particular, $A$ becomes a $H^*$-algebra. For example, a inequality of the form

$$\begin{eqnarray*}[x,y]+[y,x]\le [x,x]^{1/2}[y,y]^{1/2}+[y,y]^{1/2}[x,x]^{1/2},\quad x,y\in E, \end{eqnarray*}$$

proposed by C.P. Niculescu, does not hold for arbitrary $\mathbb{C}^*$-algebras $A$, but is proved to be true if $A$ is a $H^*$-algebra.