An operator inequality and matrix normality Original Research Article

Let A be a bounded linear operator on a Hilbert space H; denote A = (A*A)1/2 and the norm of x ε H by double vertical barxdouble vertical bar. It is proved that image(Au, v)≤triple vertical-rule fenceAaudouble vertical bar triple vertical-rule fenceA*1−adouble vertical bar for allu, v ε H for any 0 < α < 1. In particular, image(Au, v)≤(Au, u)1/2(A*v,v)1/2 for allu, v ε H. When H is of finite dimension, it is shown that A must be a normal operator if it satisfies image(Au, u)≤(Au, u)a(A*u, u)1−a for allu ε H for some real number α ≠ 1/2.