Young type inequalities for positive operators (Q1947170)

The well-known Young inequality states that, if \(a,b\geq0\) and \(0\leq\nu\leq1\), then \[ a^{\nu}b^{1-\nu}\leq \nu a+(1-\nu)b. \] It is known that, if we replace the scalars \(a,b\) with two positive operators \(A,B\in\mathbb{B}(H)\), the above inequality does not hold. However, \textit{T. Ando} [Oper. Theory, Adv. Appl. 75, 33--38 (1994; Zbl 0830.47010)] proved that \[ |||A^{\nu}X B^{1-\nu}|||\leq \nu |||AX|||+(1-\nu)|||XB||| \] for positive operators \(A,B\) and a unitarily invariant norm \(|||\cdot|||\). In Section 2, the author presents a refinement of Ando's inequality. In Section 3, he reviews some results related to the Heinz inequality [\textit{E. Heinz}, Math. Ann. 123, 415--438 (1951; Zbl 0043.32603)] and improves some of them. Furthermore, some new proofs of known results are obtained.