Finite-rank operators are matrices 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 ∈ Cn × m has rank 1if and only ifM 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 and 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 accumulate only at 0, T is then a compact operator, and one has the canonical form for compact operators. If the series ∑iαi is convergent, T is a trace class operator.
Algebraic property
The family of finite-rank operators F on a Hilbert space H form a two-sided *-ideal in L, 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 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 Sh, k that maps h to k lies in I. Define Sh, f to be the rank-1 operator that maps h to f, and Sg, k analogously. Then which means Sh, k is in I and this verifies the claim. Some examples of two-sided *-ideals in L are the trace-class, Hilbert–Schmidt operators, and compact operators. F is dense in all three of these ideals, in their respective norms. Since any two-sided ideal in L must contain F, the algebra L is simple if and only if it is finite dimensional.
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.
