{{Short description|Linear operator in functional analysis}} {{More citations needed|date=June 2021}} In [[functional analysis]], a branch of mathematics, a '''finite-rank operator''' is a [[bounded linear operator]] between [[Banach space]]s whose [[Image (mathematics)|range]] is finite-dimensional.<ref>{{cite web|url=https://www.sciencedirect.com/topics/mathematics/finite-rank-operator|title=Finite Rank Operator - an overview|date=2004}}</ref>
==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, <math> M \in \mathbb{C}^{n \times m} </math> has rank <math>1</math> [[if and only if]] <math>M</math> is of the form
:<math>M = \alpha \cdot u v^*, \quad \mbox{where} \quad \|u \| = \|v\| = 1 \quad \mbox{and} \quad \alpha \geq 0 .</math>
Exactly the same argument shows that an operator <math>T</math> on a [[Hilbert space]] <math>H</math> is of rank <math>1</math> if and only if
:<math>T h = \alpha \langle h, v\rangle u \quad \mbox{for all} \quad h \in H ,</math>
where the conditions on <math> \alpha, u, v </math> are the same as in the finite dimensional case.
Therefore, by induction, an operator <math>T</math> of finite rank <math>n</math> takes the form
:<math>T h = \sum _{i = 1} ^n \alpha_i \langle h, v_i\rangle u_i \quad \mbox{for all} \quad h \in H ,</math>
where <math>\{ u_i \}</math> and <math>\{v_i\}</math> 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 <math>n</math> is now countably infinite and the sequence of positive numbers <math>\{ \alpha_i \} </math> [[limit point|accumulate]] only at <math>0</math>, <math>T</math> is then a [[compact operator on Hilbert space|compact operator]], and one has the canonical form for compact operators.
Compact operators are [[trace class]] only if the series <math display="inline"> \sum _i \alpha _i </math> is convergent; a property that automatically holds for all finite-rank operators.<ref>{{cite book|last=Conway|first=John B.|author-link=John B. Conway|title=A course in functional analysis|publisher=Springer-Verlag|publication-place=New York|year=1990|isbn=978-0-387-97245-9|oclc=21195908|pages=267–268}}</ref>
===Algebraic property=== The family of finite-rank operators <math>F(H)</math> on a Hilbert space <math>H</math> form a two-sided *-ideal in <math>L(H)</math>, the algebra of bounded operators on <math>H</math>. In fact it is the minimal element among such ideals, that is, any two-sided *-ideal <math>I</math> in <math>L(H)</math> must contain the finite-rank operators. This is not hard to prove. Take a non-zero operator <math>T\in I</math>, then <math>Tf = g</math> for some <math>f, g \neq 0</math>. It suffices to have that for any <math>h, k\in H</math>, the rank-1 operator <math> S_{h, k} </math> that maps <math>h</math> to <math>k</math> lies in <math>I</math>. Define <math> S_{h, f} </math> to be the rank-1 operator that maps <math>h</math> to <math>f</math>, and <math> S_{g,k}</math> analogously. Then
:<math>S_{h,k} = S_{g,k} T S_{h,f}, \,</math>
which means <math> S_{h, k} </math> is in <math>I</math> and this verifies the claim.
Some examples of two-sided *-ideals in <math> L(H) </math> are the [[trace-class]], [[Hilbert–Schmidt operator]]s, and [[compact operator]]s. <math> F(H)</math> is dense in all three of these ideals, in their respective norms.
Since any two-sided ideal in <math> L(H)</math> must contain <math> F(H)</math>, the algebra <math> L(H)</math> is [[simple algebra|simple]] if and only if it is finite dimensional.
==Finite-rank operators on a Banach space== A finite-rank operator <math>T:U\to V</math> between [[Banach space]]s is a [[bounded operator]] such that its [[Range of a function|range]] is finite dimensional. Just as in the Hilbert space case, it can be written in the form
:<math>T h = \sum _{i = 1} ^n \langle u_i, h\rangle v_i \quad \mbox{for all} \quad h \in U ,</math>
where now <math>v_i\in V</math>, and <math>u_i\in U'</math> are bounded linear functionals on the space <math>U</math>.
A bounded linear functional is a particular case of a finite-rank operator, namely of rank one.
==References== {{reflist}}
{{DEFAULTSORT:Finite Rank Operator}} [[Category:Operator theory]]