Hölder-type inequalities involving unitarily invariant norms
The author proves that, if \(A, B\) and \(X\) are operators acting on a complex Hilbert space, then \[ \left| \left| \left| {} \left| A^{\ast }XB\right|^{r} \right| \right| \right| ^{2}\leq \left| \left| \left| \left( A^{\ast }\left| X^{\ast} \right| A\right) ^{\frac{ pr}{2}} \right| \right| \right| ^{\frac{1}{p}} \left| \left| \left| \left( B^{\ast }\left| X\right| B\right) ^{ \frac{qr}{2}} \right| \right| \right| ^{\frac{1}{q}} \] for all positive real numbers \(r, p\) and \(q\) such that \(p ^{ - 1} + q ^{ - 1} = 1\) and for all unitarily invariant norms. This extends some inequalities of \textit{R. Bhatia} and \textit{C. A. Davis} [Linear Algebra Appl. 223--224, 119--129 (1995; Zbl 0824.47006)] and \textit{R. A. Horn} and \textit{X.-Z. Zhan} [Linear Algebra Appl. 291, No. 1--3, 103--113 (1999; Zbl 0932.15015)]. The author also obtains general Hölder-type inequalities involving unitarily invariant norms for sums and products of Hilbert space operators by utilizing block matrices of operators, majorization of the singular values, and other classical inequalities. Relations between these inequalities and the different forms of the Cauchy-Schwarz inequalities for operators are also obtained.
scientific article; zbMATH DE number 6121581