Inequalities for sums and direct sums of Hilbert space operators (Q884410)

The authors investigate several singular value inequalities and norm inequalities involving sums and direct sums of operators. They prove the following nice Cauchy--Schwarz type inequality for singular values: \[ s_j \left(\sum_{i=1}^n A_iX_iB_i\right) \leq \left\| \sum_{i=1}^n| A_i^*| ^2\right\| ^{1/2} \left\| \sum_{i=1}^n| B_i| ^2\right\| ^{1/2} s_j\left(\oplus_{i=1}^n X_i\right),\quad j=1, 2, \dots, \] where \(A_i, B_i, X_i\), \(1\leq i\leq n\), are bounded linear operators on a Hilbert space and the operators \(X_i\), \(1\leq i\leq n\), are compact. They also show that if \(X\) and \(Y\) are compact operators on a Hilbert space, then \(s_j\left(\frac{X + Y}{2}\right)\leq s_j(X \oplus Y)\), \(j=1, 2, \dots\). The authors generalize the inequality \[ \left|\left|\left|\left(\frac{X+Y}{2}\right)\oplus\left(\frac{X+Y}{2}\right)\right|\right|\right|\leq||| X\oplus Y|||, \] where \(|||\cdot|||\) is a unitarily invariant norm and \(X\) and \(Y\) are bounded linear operators on a Hilbert space by giving a weighted mean norm inequality.