RADIUS FOR HILBERT C*-MODULE OPERATORS

In this paper, we introduce some inequalities between the operator norm and the numerical radius of adjointable operators on Hilbert C*-module spaces. Moreover, we establish some new refinements of numerical radius inequalities for Hilbert space operators. More precisely, we prove that if T is an element of B (H) and min (parallel to T + T*parallel to(2)/2, parallel to T - T*parallel to(2)/2 <= max (inf(parallel to x parallel to=1) parallel to Tx parallel to(2), inf(parallel to x parallel to)=1 parallel to T*x parallel to(2)), then parallel to T parallel to <= root 2 omega(T); this is a considerable improvement of the classical inequality parallel to T parallel to <= 2 omega(T