Let $H$ be a Hilbert space and $b$ a continuous and symmetric bilinear form on $H \times H$, such that the induced operator $T \colon H \to H^{\ast}$ (the star denoting the continuous dual) is Hilbert-Schmidt. Suppose that $H$ is continuously and densely embedded in a Banach space $B$ and that

$$ \sup_{h_1,h_2 \in H \setminus \left\{ 0 \right\}} \frac{ \left| b(h_1,h_2) \right| }{ \left\lVert h_1 \right\rVert_H \left\lVert h_2 \right\rVert_B }\leq C $$

where the norms on the right are those associated to $H$ and $B$. Then $T$ can be extended to a continuous linear operator $T_1 \colon B \to H^{\ast}$. Also, if I'm not mistaken, the above inequality allows to consider $T$ as a continuous linear operator $T_2 \colon H \to B^{\ast}$.

**My question**: Can the 2-summing norm $\pi_2(T_2)$ of $T_2$ be estimated by the Hilbert-Schmidt norm $\left\lVert T \right\rVert_2$ of $T$, i.e. does there exist some (ideally explicit) $\varphi$ such that $\pi_2(T_2) \leq \varphi(|\left\lVert T \right\rVert_2)$? In the answer to this question, it was mentioned that

$$ \pi_2(T_2) = \sup \left( \sum_n^{} \left\lVert T_2e_n \right\rVert_{B^{\ast}}^2 \right)^{1/2}, $$

where the supremum is taken over all orthonormal bases $\left\{ e_n \right\}$ of $H$, which might be relevant here.

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