The Schwarz inequality in a Hilbert space (with real or complex scalars) implies that: [(Ax,Bx)|2 < (Ax,Ax)(Bx,Bx), where x is an arbitrary vector of the Hilbert space to itself. This inequality may be regarded as giving an upper bound for the number |(Ax,Bx)|2 in terms of the two "squares of the norms" (Ax,Ax) and (Bx,Bx).nUnder certain circumstances (i.e., additional suitable hypotheses relative to the operators A and B) there exist complementary inequalities to the Schwarz inequality; that is to say. inequalities which can be regarded as giving lower bounds for |(Ax,Bx)| in terms of (Ax,Ax) and (Bx,Bx). The purpose of the present paper is to prove operator inequalities of this general "complementary" nature.
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