Hankel Forms and Embedding Theorems in Weighted Dirichlet Spaces

We show that for a fixed operator-valued analytic function g the boundedness of the bilinear (Hankel-type) form $(f,\ h)\rightarrow \int_{\mathbb D}\ {\mathrm tr}(g'^*\ fh')(1-|z|^2)^\alpha\ {\mathrm d}A$, defined on appropriate cartesian products of dual weighted Dirichlet spaces of Schatten class-valued functions, is equivalent to corresponding Carleson embedding estimates.