# Finite-rank operator

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In functional analysis, a branch of mathematics, a **finite-rank operator** is a bounded linear operator between Banach spaces whose range is finite-dimensional.

## 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} has rank 1 if and only if *M* is of the form

Exactly the same argument shows that an operator *T* on a Hilbert space *H* is of rank 1 if and only if

where the conditions on *α*, *u*, and *v* are the same as in the finite dimensional case.

Therefore, by induction, an operator *T* of finite rank *n* takes the form

where {*u _{i}*} and {

*v*} are orthonormal bases. Notice this is essentially a restatement of singular value decomposition. This can be said to be a

_{i}*canonical form*of finite-rank operators.

Generalizing slightly, if *n* is now countably infinite and the sequence of positive numbers {*α _{i}*} accumulate only at 0,

*T*is then a compact operator, and one has the canonical form for compact operators.

If the series ∑_{i} *α _{i}* is convergent,

*T*is a trace class operator.

### Algebraic property

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

which means *S*_{h, k} is in *I* and this verifies the claim.

Some examples of two-sided *-ideals in *L*(*H*) are the trace-class, Hilbert–Schmidt operators, and compact operators. *F*(*H*) is dense in all three of these ideals, in their respective norms.

Since any two-sided ideal in *L*(*H*) must contain *F*(*H*), the algebra *L*(*H*) is simple if and only if it is finite dimensional.

## Finite-rank operators on a Banach space

A finite-rank operator 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

where now , and are bounded linear functionals on the space .

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