Some inequalities for adjointable operators on Hilbert \(C^\ast\)-modules (Q6622546)

The authors give a new characterization of the positivity of \(2\times 2\) operator matrices on Hilbert \(C^{\ast }\)-modules: Let \(H\) and \(K\) be two Hilbert \(C^{\ast }\)-modules over \(\mathcal{A}\). Then \N\(\left[ \begin{array}{cc} A & C \\ C^{\ast } & B \end{array} \right] \) is positive in \(L(H\oplus K)\) if and only if \(A\) and \(B\) are positive and \(\left\Vert \left\langle Cx,y\right\rangle \right\Vert ^{2}\leq \left\Vert \left\langle Ax,x\right\rangle \right\Vert \left\Vert \left\langle By,y\right\rangle \right\Vert \) for all \(x\in H\) and \(y\in K.\) Also, they present a generalized version of the mixed Schwarz inequality in the context of Hilbert \(C^{\ast }\)-modules.