Interpolation unitarily invariant norms inequalities for matrices with applications

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:

$ \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*} $

where

$ \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*} $

and

$ \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*} $

Many other related interpolation inequalities are also obtained.