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 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,
is of the form The same argument and Riesz' lemma show that an operator
are the same as in the finite dimensional case.
Therefore, by induction, an operator
takes the form where
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
is now countably infinite and the sequence of positive numbers
is then a compact operator, and one has the canonical form for compact operators.
Compact operators are trace class only if the series
is convergent; a property that automatically holds for all finite-rank operators.
[2] The family of finite-rank operators
form a two-sided *-ideal in
, the algebra of bounded operators on
In fact it is the minimal element among such ideals, that is, any two-sided *-ideal
must contain the finite-rank operators.
Take a non-zero operator
to be the rank-1 operator that maps
is dense in all three of these ideals, in their respective norms.
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
are bounded linear functionals on the space
A bounded linear functional is a particular case of a finite-rank operator, namely of rank one.