Definition:Finite Rank Operator
Let $H, K$ be Hilbert spaces.
Let $T: H \to K$ be a linear transformation.
Note that a finite rank operator is not necessarily bounded.
As linear operator usually has a more specified meaning, the name finite rank operator is a case of 'pars pro toto'.
This might be due to finite rank linear transformation, which is formally more correct, being awkward in pronunciation.