Operators on a Tensor Product Space Suppose $V$ and $W$ are vector spaces over the same field.  Is $\text{End}(V \otimes W)$ the same as $\text{End}(V) \otimes \text{End}(W)$?  Are there special names for these spaces?

Julian's proof only works in the finite-dimensional case (dual bases etc.).
More generally, for $R$-modules $M,N,M',N'$ there is a canonical $R$-linear map $\hom(M,M') \otimes \hom(N,N') \to \hom(M \otimes N,M' \otimes N')$, which is an isomorphism when $M$ and $N$ are finitely generated projective (since it holds for $M=R$ and the $M$s satisfying this are closed under finite direct sums and direct summands; see also here). In particular, $\mathrm{End}(M) \otimes \mathrm{End}(N) \to \mathrm{End}(M \otimes N)$ is an isomorphism.
If $V,W$ are infinite-dimesional vector spaces, then the canonical homomorphism $\mathrm{End}(V) \otimes \mathrm{End}(W) \to \mathrm{End}(V \otimes W)$ is still injective, but not surjective. See my answer here.
