A remark on interpolation in spaces of vector functions

Let B(H) be the space of bounded operators in a Hilbert space $H$, let $B_p^s(\gamma_p)$ be the Besov class of functions, analytic in the unit circle $\mathbb D$ and taking values in the Schatten–von Neumann class $\gamma_p(H)$, and let $X=\mathbb P_+L^{\infty}(B(H))=\{\sum_{n\ge0}\hat{f}(n)z^n:f\in L^{\infty}(B(H))\}$. The fundamental result is that $(B_p^{1/p}(\gamma_p),X)_{\theta,q}=B_q^{1/q}(\gamma_q),\quad 1\le p<\infty,\quad 0<\theta<1,\quad q=\dfrac{p}{1-\theta}$.