Rank-one operator - Knowlpedia

Let $X$ and $Y$ be normed vector spaces over the same field, $\mathbb R$ or $\mathbb C$ . A rank-one operator is a bounded linear operator $T:X\to Y$ such that $\dim T(X)=1$ .

Representation

Every rank-one operator can be written

Tx=\varphi(x)y,

where $y\in Y$ is nonzero and $\varphi:X\to\mathbb F$ is a nonzero bounded linear functional. Conversely, every such formula defines a rank-one operator. Its range is the line spanned by $y$ .

Scope

The zero operator has rank zero and is therefore not rank one.

Rank one here refers to the dimension of the linear range. In Hilbert $C^*$ -module theory, a “rank-one operator” can instead mean $\theta_{y,x}(z)=y\langle x,z\rangle$ ; that Hilbert-module convention is defined by compact operators on Hilbert modules and need not have one-dimensional range.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990. Springer DOI record. Relevant: Chapter II, finite-rank operators.