If $E,F,X,Y$ are $\mathbb R$-Banach spaces, $S\in\mathfrak L(X,E)$ and $T\in\mathfrak L(Y,F)$, there is a unique bounded linear operator $S\otimes_\pi T$ from $X\:\hat\otimes_\pi\:Y$ to $E\:\hat\otimes_\pi\:F$ with $$(S\otimes_\pi T)(x\otimes y)=Sx\otimes Ty\;\;\;\text{for all }(x,y)\in X\times Y.\tag2$$ Since $$B:\mathfrak L(X,E)\times\mathfrak L(Y,F)\to\mathfrak L(X\:\hat\otimes_\pi\:Y,E\:\hat\otimes_\pi\:F)\;,\;\;\;(S,T)\mapsto S\otimes_\pi T$$ is bilinear, there is a unique bounded linear operator $$\tilde B:\mathfrak L(X,E)\:\hat\otimes_\pi\:\mathfrak L(Y,F)\to\mathfrak L(X\:\hat\otimes_\pi\:Y,E\:\hat\otimes_\pi\:F)$$ with $$\tilde B(S\otimes T)=B(S,T)\tag2\;\;\;\text{for all }(S,T)\in\mathfrak L(X,E)\times\mathfrak L(Y,F).$$
Is $\tilde B$ surjective? Or even an isometric isomorphism?