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:

$${\left|\left|\left|\sum _{j=1}^m{A}_j{P}_j{Q_j}^{\ast }{B_j}^{\ast }\right|\right|\right|}^2\le {\left(max\left\{L,M\right\}\right)}^4\left|\left|\left|K_c\right|\right|\right|\left|\left|\left|K_{1-c}\right|\right|\right|,$$

where

$$L={\left|\left|\sum _{j=1}^m{\left|{A_j}^{\ast }\right|}^2\right|\right|}^{\frac{1}{2}},M={\left|\left|\sum _{j=1}^m{\left|{B_j}^{\ast }\right|}^2\right|\right|}^{\frac{1}{2}},$$

and

$$K_c=\left(c{\left|P_1\right|}^2+\left(1-c\right){\left|Q_1\right|}^2\right)\oplus \cdots \oplus \left(c{\left|P_m\right|}^2+\left(1-c\right){\left|Q_m\right|}^2\right).$$

Many other related interpolation inequalities are also obtained.