Definition:Finite Rank Operator

Definition
Let $H, K$ be Hilbert spaces.

Let $T: H \to K$ be a linear transformation.

Then $T$ is said to be a finite rank operator, or of finite rank, iff its range, $\operatorname{ran} T$, is finite dimensional.

Note that a finite rank operator is not necessarily bounded.

Note
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.