Generalizations of the unitarily invariant norm version of the arithmetic–geometric mean inequality

<abstract>Let $ A_j, B_j, P_j $, and $ Q_j \in M_{n}(\mathbb{C}) $, where $ j = 1, 2, \dots, m $. For a real number $ c \in [0, 1] $, we prove the following interpolation inequality: <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} {\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert {\sum\limits_{j = 1}^m {{A_j}{P_j}{Q_j}^*{B_j}^*} } \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert}^2 \leq {\left( {\max \left\{ {L,\,M} \right\}} \right)^4} {\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert {K_c} \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert} \quad {\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert {K_{1-c}} \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert}, \end{equation*} $\end{document} </tex-math></disp-formula> where <disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} L = {\left\vert\kern-0.1ex\left\vert {\sum\limits_{j = 1}^m {\left| {{A_j}^*} \right|^2} } \right\vert\kern-0.1ex\right\vert}^ \frac{1}{2}, M = {\left\vert\kern-0.1ex\left\vert {\sum\limits_{j = 1}^m {\left| {{B_j}^*} \right|^2} } \right\vert\kern-0.1ex\right\vert}^ \frac{1}{2}, \end{equation*} $\end{document} </tex-math></disp-formula> and <disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{equation*} K_c = \left( {c{{\left| {{P_1}} \right|}^2} + \left( {1 - c} \right){{\left| {{Q_1}} \right|}^2}} \right) \oplus \cdots \oplus \left( {c{{\left| {{P_m}} \right|}^2} + \left( {1 - c} \right){{\left| {{Q_m}} \right|}^2}} \right). \end{equation*} $\end{document} </tex-math></disp-formula> Many other related interpolation inequalities are also obtained.</abstract>

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Generalizations of the unitarily invariant norm version of the arithmetic–geometric mean inequality

Cited by 2 publications

References 10 publications

Operator inequalities related to the arithmetic–geometric mean inequality and characterizations

Operator inequalities related to the arithmetic–geometric mean inequality and characterizations

Interpolation unitarily invariant norms inequalities for matrices with applications

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