Finite-rank operator

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 \mathbb{C}^{n \times m} has rank 1 if and only if M is of the form

:M = \alpha \cdot u v^*, \quad \mbox{where} \quad |u | = |v| = 1 \quad \mbox{and} \quad \alpha \geq 0 .

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

:T h = \alpha \langle h, v\rangle u \quad \mbox{for all} \quad 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 form

:T h = \sum _{i = 1} ^n \alpha_i \langle h, v_i\rangle u_i \quad \mbox{for all} \quad 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 \sum _i \alpha _i is convergent; a property that automatically holds for all finite-rank operators.

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 \neq 0. It suffices to have that for any h, k\in 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

:S_{h,k} = S_{g,k} T S_{h,f}, ,

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 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 = \sum _{i = 1} ^n \langle u_i, h\rangle v_i \quad \mbox{for all} \quad 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.

References

References

1. (2004). "Finite Rank Operator - an overview".
2. Conway, John B.. (1990). "A course in functional analysis". Springer-Verlag.