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, if and only if its range, $\Rng T$, is finite dimensional.

Note that a finite rank operator is not necessarily bounded.

Linguistic Note

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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

• Space of Continuous Finite Rank Operators

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $II.4.3$