Some vector inequalities for two operators in Hilbert spaces with applications

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Abstract

In this paper we establish some vector inequalities for two operators related to Schwarz and Buzano results. We show amongst others that in a Hilbert space H we have the inequality

12[|A|2+|B|22x,x1/2|A|2+|B|22y,y1/2+||A|2+|B|22x,y|]|Re(B*A)x,y|
for A, B two bounded linear operators on H such that Re (B*A) is a nonnegative operator and any vectors x, yH.

Applications for norm and numerical radius inequalities are given as well.

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