Operator-valued inner product and operator inequalities
For a Hilbert space \(H\), the \(C^*\)-inner product \({\mathbf Y}^*{\mathbf X}=\sum_{j=1}^nY_j^*X_j\) \(({\mathbf X}=(X_j)\), \({\mathbf Y}=(Y_j) \in B(H,H^n))\) makes \(B(H,H^n)\) into a Hilbert \(C^*\)-module over \(B(H)\). Using this operator-valued inner product, the author of present paper nicely presents operator versions for the Schwarz and Jensen inequalities and gives simple conditions that the equalities hold. Among his results, we mention the following version of the Schwarz inequality: for operators \(X\) and \(Y\) acting on a Hilbert space, the inequality \[ Y^*X^*YX+X^*Y^*XY\leq (XY)^*XY+(YX)^*YX \] holds with equality only when \(X\) commutes with \(Y\).
scientific article; zbMATH DE number 5279269