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Young type inequalities for positive operators. (English) Zbl 1279.47031
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, 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 [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.

47A63 Linear operator inequalities
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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