Finite-rank operator explained

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

From linear algebra, we know that a rectangular matrix, with complex entries,

M \in C ^{n} has rank 1 if and only if M is of the formM = \alpha ⋅ u v ^{*,} where \|u \| = \|v\| = 1 and \alpha \geq 0 .

Exactly the same argument shows that an operator

T on a Hilbert space H is of rank 1 if and only ifT h = \alpha \langle h, v\rangle u forall h \in H ,

where the conditions on

\alpha, u, v are the same as in the finite dimensional case.

Therefore, by induction, an operator

T of finite rank n takes the formT h = \sum _{i} ^{n} \alpha _{i} \langle h, v _{i\rangle} u _{i} forall h \in H ,

where

\{ u _{i} \} and \{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 is now countably infinite and the sequence of positive numbers \{ \alpha _{i} \} accumulate only at 0 , T is then a compact operator, and one has the canonical form for compact operators.

Compact operators are trace class only if the series $\backslash sum\; \_i\; \backslash alpha\; \_i$ is convergent; a property that automatically holds for all finite-rank operators.^[2]

Algebraic property

The family of finite-rank operators

F(H) 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\in I , then Tf = g for some f, g ≠ 0 . It suffices to have that for any h, k\in H , the rank-1 operator S _{h,} that maps h to k lies in I . Define S _{h,} to be the rank-1 operator that maps h to f , and S _{g,k} analogously. ThenS _{h,k} = S _{g,k} T S _{h,f} ,

which means

S _{h,} 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

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 formT h = \sum _{i} ^{n} \langle u _{i,} h\rangle v _{i} forall h \in U ,

where now

v _{i\in} V , and u _{i\in} U' are bounded linear functionals on the space U .

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

Notes and References

1. Web site: Finite Rank Operator - an overview. 2004.
2. Book: Conway, John B.. A course in functional analysis. Springer-Verlag. New York. 1990. 978-0-387-97245-9. 21195908. 267–268.