# Finite-rank operator

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Linear operator in functional analysis

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In [functional analysis](/source/Functional_analysis), a branch of mathematics, a **finite-rank operator** is a [bounded linear operator](/source/Bounded_linear_operator) between [Banach spaces](/source/Banach_space) whose [range](/source/Image_(mathematics)) is finite-dimensional.[1]

## Finite-rank operators on a Hilbert space

### A canonical form

Finite-rank operators are matrices (of finite size) transplanted to the infinite dimensional setting. As such, these operators may be described via [linear algebra](/source/Linear_algebra) techniques.

From linear algebra, we know that a rectangular matrix, with complex entries, M ∈ C n × m {\displaystyle M\in \mathbb {C} ^{n\times m}} has rank 1 {\displaystyle 1} [if and only if](/source/If_and_only_if) M {\displaystyle M} is of the form

- M = α ⋅ u v ∗ , where ‖ u ‖ = ‖ v ‖ = 1 and α ≥ 0. {\displaystyle M=\alpha \cdot uv^{*},\quad {\mbox{where}}\quad \|u\|=\|v\|=1\quad {\mbox{and}}\quad \alpha \geq 0.}

Exactly the same argument shows that an operator T {\displaystyle T} on a [Hilbert space](/source/Hilbert_space) H {\displaystyle H} is of rank 1 {\displaystyle 1} if and only if

- T h = α ⟨ h , v ⟩ u for all h ∈ H , {\displaystyle Th=\alpha \langle h,v\rangle u\quad {\mbox{for all}}\quad h\in H,}

where the conditions on α , u , v {\displaystyle \alpha ,u,v} are the same as in the finite dimensional case.

Therefore, by induction, an operator T {\displaystyle T} of finite rank n {\displaystyle n} takes the form

- T h = ∑ i = 1 n α i ⟨ h , v i ⟩ u i for all h ∈ H , {\displaystyle Th=\sum _{i=1}^{n}\alpha _{i}\langle h,v_{i}\rangle u_{i}\quad {\mbox{for all}}\quad h\in H,}

where { u i } {\displaystyle \{u_{i}\}} and { v i } {\displaystyle \{v_{i}\}} are orthonormal bases. Notice this is essentially a restatement of [singular value decomposition](/source/Singular_value_decomposition). This can be said to be a *canonical form* of finite-rank operators.

Generalizing slightly, if n {\displaystyle n} is now countably infinite and the sequence of positive numbers { α i } {\displaystyle \{\alpha _{i}\}} [accumulate](/source/Limit_point) only at 0 {\displaystyle 0} , T {\displaystyle T} is then a [compact operator](/source/Compact_operator_on_Hilbert_space), and one has the canonical form for compact operators.

Compact operators are [trace class](/source/Trace_class) only if the series ∑ i α i {\textstyle \sum _{i}\alpha _{i}} is convergent; a property that automatically holds for all finite-rank operators.[2]

### Algebraic property

The family of finite-rank operators F ( H ) {\displaystyle F(H)} on a Hilbert space H {\displaystyle H} form a two-sided *-ideal in L ( H ) {\displaystyle L(H)} , the algebra of bounded operators on H {\displaystyle H} . In fact it is the minimal element among such ideals, that is, any two-sided *-ideal I {\displaystyle I} in L ( H ) {\displaystyle L(H)} must contain the finite-rank operators. This is not hard to prove. Take a non-zero operator T ∈ I {\displaystyle T\in I} , then T f = g {\displaystyle Tf=g} for some f , g ≠ 0 {\displaystyle f,g\neq 0} . It suffices to have that for any h , k ∈ H {\displaystyle h,k\in H} , the rank-1 operator S h , k {\displaystyle S_{h,k}} that maps h {\displaystyle h} to k {\displaystyle k} lies in I {\displaystyle I} . Define S h , f {\displaystyle S_{h,f}} to be the rank-1 operator that maps h {\displaystyle h} to f {\displaystyle f} , and S g , k {\displaystyle S_{g,k}} analogously. Then

- S h , k = S g , k T S h , f , {\displaystyle S_{h,k}=S_{g,k}TS_{h,f},\,}

which means S h , k {\displaystyle S_{h,k}} is in I {\displaystyle I} and this verifies the claim.

Some examples of two-sided *-ideals in L ( H ) {\displaystyle L(H)} are the [trace-class](/source/Trace-class), [Hilbert–Schmidt operators](/source/Hilbert%E2%80%93Schmidt_operator), and [compact operators](/source/Compact_operator). F ( H ) {\displaystyle F(H)} is dense in all three of these ideals, in their respective norms.

Since any two-sided ideal in L ( H ) {\displaystyle L(H)} must contain F ( H ) {\displaystyle F(H)} , the algebra L ( H ) {\displaystyle L(H)} is [simple](/source/Simple_algebra) if and only if it is finite dimensional.

## Finite-rank operators on a Banach space

A finite-rank operator T : U → V {\displaystyle T:U\to V} between [Banach spaces](/source/Banach_space) is a [bounded operator](/source/Bounded_operator) such that its [range](/source/Range_of_a_function) is finite dimensional. Just as in the Hilbert space case, it can be written in the form

- T h = ∑ i = 1 n ⟨ u i , h ⟩ v i for all h ∈ U , {\displaystyle Th=\sum _{i=1}^{n}\langle u_{i},h\rangle v_{i}\quad {\mbox{for all}}\quad h\in U,}

where now v i ∈ V {\displaystyle v_{i}\in V} , and u i ∈ U ′ {\displaystyle u_{i}\in U'} are bounded linear functionals on the space U {\displaystyle U} .

A bounded linear functional is a particular case of a finite-rank operator, namely of rank one.

## References

1. **[^](#cite_ref-1)** ["Finite Rank Operator - an overview"](https://www.sciencedirect.com/topics/mathematics/finite-rank-operator). 2004.

1. **[^](#cite_ref-2)** [Conway, John B.](/source/John_B._Conway) (1990). *A course in functional analysis*. New York: Springer-Verlag. pp. 267–268. [ISBN](/source/ISBN_(identifier)) [978-0-387-97245-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-97245-9). [OCLC](/source/OCLC_(identifier)) [21195908](https://search.worldcat.org/oclc/21195908).

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