Improved Young and Heinz operator inequalities for unitarily invariant norms

Year, pages: 2020, 453 - 466

In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm $|||·|||$ and $A,B,X\in M_n(\mathbb{C})$: \begin{align*} \Big|\Big|\Big|((A^ν XB^1-ν+A^1-νXB^ν) / (2))\Big|\Big|\Big| ≤ &(4r₀-1)|||A^{((1) / (2))}XB^{((1) / (2))}||| &+2(1-2r₀)\Big|\Big|\Big|(1-α)A^{((1) / (2))}XB^{((1) / (2))} +α\Big(((AX+XB) / (2))\Big)\Big|\Big|\Big|, \end{align*} where $((1) / (4))≤ ν ≤ ((3) / (4))$, $α \in [((1) / (2)),∞ )$ and $r₀=\min\{ν,1-ν\}$.

Beiranvand, A., Ghazanfari, A. 2020. Improved Young and Heinz operator inequalities for unitarily invariant norms. In Mathematica Slovaca, vol. 70, no.2, pp. 453-466. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0363

Beiranvand, A., Ghazanfari, A. (2020). Improved Young and Heinz operator inequalities for unitarily invariant norms. Mathematica Slovaca, 70(2), 453-466. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0363

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava