Let $H$ denote a separable Hilbert space, $E$ a separable Banach space, $E'$ the dual space of $E$ and $E''$ the bidual of $E$. Let $ I: H \to E$ be a bounded operator and $I^*:E' \to H$ denote the adjoint operator of $I$.
Assuming that $S := II^* :E' \to E$ is a nuclear operator meaning $S e'= \sum_{j=1}^\infty \lambda_j \langle e', e_j'' \rangle e_j$, also $e_j'' \in E''$ and $e_j \in E$ with $\Vert e_j'' \Vert_{E''} = \Vert e_j \Vert_E =1$ and $\sum_{j=1}^\infty |\lambda_j| < \infty $.
Can one conclude that the Operator $I : H \to E$ satisfies $\sum_{j=1}^\infty \Vert I h_j \Vert_E^2 < \infty $ for any orthonormal basis $(h_j)_{j=1}^\infty \subset H$?