Let $\{H_i:i\in I\}$ be a system of complex Hilbert spaces, $L(\bigoplus_{i\in I} H_i)$ the set of bounded linear maps $T:\bigoplus_{i\in I} H_i\to \bigoplus_{i\in I} H_i$, equipped with the operator norm. Let $\prod_{i\in I}L(H_i)$ the direct product of linear maps endowed with the norm $\|(T_i)_{i\in I}\|=sup_{i\in I}\|T_i\|$.
Is $$L(\bigoplus_{i\in I} H_i)\cong \prod_{i\in I}L(H_i)$$ as banach spaces? The map which comes to my mind constructed as follows: Is $$\iota_i: H_i\to \bigoplus_{i\in I} H_i$$ the canonical injection, we define for each $i\in I$ a map $$\varphi_i:L(\bigoplus_{i\in I} H_i)\to L(H_i),\; \alpha\mapsto \alpha\circ \iota_i$$ Then extend it to a map $$L(\bigoplus_{i\in I} H_i)\to \prod_{i\in I}L(H_i).$$ But I'm not sure if it's possible here or if the construction makes sense in this context.
Note that isomorphically there is only one Hilbert space in each dimension. Moreover, $$d( \bigoplus_{i\in I} H_i)=\sum_{i\in I} d(H_i),$$ where $d$ stands for the dimension, that is, cardinality of any orthonormal basis. Thus if all $H_i$ have the same dimension $\lambda$ say, then $d(\bigoplus_{i\in I} H_i) = \max\{\lambda, |I|\}$. Presumably, by $\prod_{i\in I} E_i$ you mean the $\ell_\infty(I)$-direct sum of $E_i$ ($i\in I$).
In general then the answer is no.
Indeed, if each $H_i$ is two-dimensional and $I$ is infinite, then $(\bigoplus_{i\in I} L(H_i))_{\ell_\infty(I)}$ is Banach-space isomorphic to $\ell_\infty(I)$ but $L( (\bigoplus_{i\in I}H_i)_{\ell_2(I)})$ is the space of operators on an infinite dimensional Hilbert space $H$ which is known to be not isomorphic to $\ell_\infty(I)$ for any infinite index set $X$. To see this, note that $L(H)$ contains a complemented subspace isomorphic to $\ell_2$. Pick a non-zero vector $x\in H$ and consider
$$H^\prime=\{x\otimes y\colon\quad y\in H\}.$$
Then $H^\prime$ is complemented and $H\cong H^\prime$. As $\ell_\infty$ has the Dunford–Pettis property, it contains no infinite-dimensional, complemented reflexive subspaces.
In some cases the answer is yes. For example, it is an unpublished result of Lindenstrauss and Haagerup which says that
$$L(\ell_2)\quad \text{and }\quad(\bigoplus_{n\in \mathbb{N}} L(\ell_2^n))_{\ell_\infty}$$
are isomorphic as Banach spaces. The proof is highly non-constructive and uses the Pełczyński decomposition method. It is explained in