Finite-rank operator - Imported Knowlpedia Content

Definition

Let $X$ and $Y$ be normed vector spaces over the same scalar field $\mathbb R$ or $\mathbb C$ . A bounded linear operator $T:X\to Y$ is a finite-rank operator if its range $T(X)$ is finite-dimensional. Its rank is

\operatorname{rank}T=\dim T(X).

Thus the zero operator has rank $0$ , and $T$ has rank $n$ precisely when its range has a Hamel basis of $n$ vectors. Boundedness is part of the operator-theoretic convention used here; an algebraic linear map can have finite-dimensional range without being continuous.

Rank-one decompositions

An operator has finite rank exactly when it can be written

Tx=\sum_{j=1}^{n}\varphi_j(x)y_j

for finitely many continuous linear functionals $\varphi_j\in X'$ and vectors $y_j\in Y$ . Choosing the $y_j$ as a basis for $T(X)$ gives such a representation, while every displayed sum has range contained in their span. Rank-one operators $x\mapsto\varphi(x)y$ are the elementary building blocks Conway, Chapter II.

Approximation and examples

Finite-rank operators form a linear subspace of $B(X,Y)$ , are compact, and are stable under composition with bounded operators. Their operator-norm closure is the class of approximable operators; it need not contain every compact operator for arbitrary Banach spaces. On Hilbert spaces, compact operators are operator-norm limits of finite-rank operators, and finite-rank operators lie in every Schatten class.

The identity on $X$ is finite-rank exactly when $X$ is finite-dimensional. A nonzero functional $\varphi\in X'$ and vector $y\in Y$ give a rank-one operator $x\mapsto\varphi(x)y$ .

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics 96, Springer, 1990. Springer DOI record. Relevant: Chapter II, “Operators on Hilbert Space,” and the discussion of finite-rank and compact operators.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980. Elsevier book record. Relevant: Chapter VI, “Bounded Operators.”