Suppose we are given Hilbert spaces $Z = X \oplus Y$ and $C = A \oplus B$. Let $T : X \rightarrow A$ be a bounded linear operator.
We can identify it with the operator $S : Z \rightarrow C$ that maps $(x,y) \in Z$ to $S(x,y) = ( Tx, 0 )$.
This is a basic construction, but how do you call it? $S$ extends the domain and the codomain of $T$. I hesitate to call this the trivial extension. I think a similar question can be imposed in other categories as well, and in differential geometry the terms push-forward and pull-back would enter for a similar construction. Can you help me in finding the proper name of $S$?