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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 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 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.}

The same argument and Riesz' lemma show that an operator T {\displaystyle T} on a 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. 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 only at 0 {\displaystyle 0} , T {\displaystyle T} is then a compact operator, and one has the canonical form for compact operators.

Compact operators are 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, Hilbert–Schmidt operators, and compact operators. 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 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 is a bounded operator such that its range 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. "Finite Rank Operator - an overview". 2004. https://www.sciencedirect.com/topics/mathematics/finite-rank-operator ↩

2. Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. pp. 267–268. ISBN 978-0-387-97245-9. OCLC 21195908. 978-0-387-97245-9 ↩