On The Interpolation of Injective or Projective Tensor Products of Banach Spaces

We prove a general result on the factorization of matrix-valued analytic functions. We deduce that if $(E_0,E_1)$ and $(F_0,F_1)$ are interpolation pairs with dense intersections, then under some conditions on the spaces $E_0$, $E_1$, $F_0$ and $F_1$, we have $$[E_0\widehat{\otimes }{F}_0,E_1\widehat{\otimes }{F}_1{]}_t=[E_0,E_1{]}_t\widehat{\otimes }[F_0,F_1{]}_t,0<t<1.$$ We find also conditions on the spaces $E_0$, $E_1$, $F_0$ and $F_1$, so that the following holds $$[E_0\text{wcheck}\otimes F_0,E_1\text{wcheck}\otimes F_1{]}_t=[E_0,E_1{]}_t\text{wcheck}\otimes [F_0,F_1{]}_t,0<t<1.$$ Some applications of these results are also considered.