On operator inequalities and linear combinations of operators

AbstractLet A1,…,Ak,C,PϵB(H), the Banach algebra of all bounded linear operators on a complex Hilbert space H. It is shown that (1) A1PA∗1+⋯+AkPA∗ k⩾CPC∗ for all P⩾0 if and only if for every x in H there exist scalars a1(x),…,ak(x) with |a1(x)|2+⋯+|ak(x)|2⩽1 such that Cx=a1(x)A1x+⋯+ak(x)Akx; (2) A1PA∗1+A2PA∗2⩾CPC∗ for all P⩾0 if and only if C=aA1 +bA2 with |a|2+|b|2 ⩽1; (3) if there is a vector x in H such that A1x,…,Akx are linearly independent, then A1PA∗1+⋯+AkPA∗k⩾CPC∗ for each P⩾0 if and only if C=a1A1 +⋯+akAk with |a1|2+⋯+|ak|2⩽1. Finally, some applications of the above results to the representation of positivity-preserving elementary operators on B(H) are given