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Cauchy-Schwarz Inequalities in Hilbert Modules

Adriana Popovici, Dan Popovici
Department of Mathematics
University of the West of Timisoara
B-dul V.Pārvan nr. 4
1900 Timisoara, Romānia
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.





Pui Ling Pang 2001-04-20