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Refinements of the Heinz inequalities for operators and matrices

In: Mathematica Slovaca, vol. 68, no. 6
Mahdi Mohammadi Gohari - Maryam Amyari
Detaily:
Rok, strany: 2018, 1431 - 1438
Kľúčové slová:
convex function, Heinz operator inequalities, Heinz mean, Hermite-Hadamard inequality, refinement, unitarily invariant norm
O článku:
Suppose that $A,B \in \mathbb{B}(\mathcal{H})$ are positive invertible operators. In this paper, we show that \begin{align*} A \# B & \leq \frac{1}{1-2\mu}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2}) A^\frac{1}{2}\\ &\leq\frac{1}{2}\bigg[A \# B +H_\mu (A,B)\bigg]\\ &\leq\frac{1}{2}\bigg[\frac{1}{1-2\mu} A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}+H_\mu (A,B)\bigg]\\ &\leq \dots \leq \frac{1}{2^n}A \# B + \frac{2^n-1}{2^n}H_\mu (A,B)\\ & \leq \frac{1}{2^n(1-2\mu)}A^\frac{1}{2}F_\mu(A^\frac{-1}{2} BA^\frac{-1}{2})A^\frac{1}{2}+\frac{2^n-1}{2^n}H_\mu (A,B)\\ & \leq \frac{1}{2^{n+1}} A \# B +\frac{2^{n+1}-1} {2^{n+1}}H_\mu (A,B)\\ &\leq \dots \leq H_\mu (A,B) \end{align*} for each $\mu \in [0,1] \smallsetminus\{\frac{1}{2}\}$, where $H_\mu (A,B)$ and $A\# B$ are the Heinz mean and the geometric mean for operators $A, B$, respectively, and $F_{\mu}\in C({\rm sp}(A^\frac{-1}{2}BA^\frac{-1}{2}))$ is a certain parameterized class of functions. As an application, we present several inequalities for unitarily invariant norms.
Ako citovať:
ISO 690:
Gohari, M., Amyari, M. 2018. Refinements of the Heinz inequalities for operators and matrices. In Mathematica Slovaca, vol. 68, no.6, pp. 1431-1438. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0192

APA:
Gohari, M., Amyari, M. (2018). Refinements of the Heinz inequalities for operators and matrices. Mathematica Slovaca, 68(6), 1431-1438. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0192
O vydaní:
Publikované: 3. 12. 2018