Hölder and Young inequalities for the trace of operators (Q885063)

The author proves the following result (of which the case \(n=2\) is well-known): given positive operators \(a_1,\ldots,a_n\) in a semifinite von Neumann algebra with trace \(\tau\), and \(p_1,\ldots,p_n>0\) with \(\frac1{p_1}+\cdots+\frac1{p_n}=1\), the following inequalities hold: \[ \tau(| a_1\cdots a_n| )\leq \prod_{j=1}^n (\tau(a_j^{p_j}))^{\frac1{p_j}}\leq\tau \left(\sum_{j=1}^n\;\frac1{p_j} a_j^{p_j}\right), \] with the case of equality characterized by \(a_1^{p_1}=\cdots=a_n^{p_n}\). The theorem is proven for matrices in Section 1, and then the extension to von Neumann algebras is argued in Section 2.