Definition:Finite Rank Operator

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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.


Also see


Sources