Image of the tensor product of strict maps of Banach spaces

Question

Let $f:A\to C$ and $g:B\to D$ be bounded linear maps of Banach spaces with closed image. Will $f\widehat{\otimes}g:A\widehat{\otimes}_\pi B\to C\widehat{\otimes}_\pi D$ also have closed image? What about injective tensor products?

Answer 1

As for the first question let $A$ be a closed subspace of $C$ and $B$ be a closed subspace of $D$. Let $f$ and $g$ be natural embeddings. Then OP's question is equivalent to the following: does projective tensor product respect embeddings? It is not, see Tensor Norms and Operator Ideals. A. Defant, K. Floret page 36, section 3.9

Now we move to the second question. Let $C$ be a quotient of $A$, and $B$ be a quotient of $D$. Assume $f$ and $g$ are respective quotient maps. Then OP's question may be formulated as follows: does injective tensor product respect quotients? Again it is not, see the same book page 49, section 4.3