Finite-rank operator

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

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

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

${\displaystyle Th=\alpha \langle h,v\rangle u\quad {\mbox{for all}}\quad h\in H,}${\displaystyle Th=\alpha \langle h,v\rangle u\quad {\mbox{for all}}\quad h\in H,}

where the conditions on ${\displaystyle \alpha ,u,v}${\displaystyle \alpha ,u,v} are the same as in the finite dimensional case.

Therefore, by induction, an operator ${\displaystyle T}${\displaystyle T} of finite rank ${\displaystyle n}${\displaystyle n} takes the form

${\displaystyle Th=\sum _{i=1}^{n}\alpha _{i}\langle h,v_{i}\rangle u_{i}\quad {\mbox{for all}}\quad h\in H,}${\displaystyle Th=\sum _{i=1}^{n}\alpha _{i}\langle h,v_{i}\rangle u_{i}\quad {\mbox{for all}}\quad h\in H,}

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

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

The family of finite-rank operators ${\displaystyle F(H)}${\displaystyle F(H)} on a Hilbert space ${\displaystyle H}${\displaystyle H} form a two-sided *-ideal in ${\displaystyle L(H)}${\displaystyle L(H)}, the algebra of bounded operators on ${\displaystyle H}${\displaystyle H}. In fact it is the minimal element among such ideals, that is, any two-sided *-ideal ${\displaystyle I}${\displaystyle I} in ${\displaystyle L(H)}${\displaystyle L(H)} must contain the finite-rank operators. This is not hard to prove. Take a non-zero operator ${\displaystyle T\in I}${\displaystyle T\in I}, then ${\displaystyle Tf=g}${\displaystyle Tf=g} for some ${\displaystyle f,g\neq 0}${\displaystyle f,g\neq 0}. It suffices to have that for any ${\displaystyle h,k\in H}${\displaystyle h,k\in H}, the rank-1 operator ${\displaystyle S_{h,k}}${\displaystyle S_{h,k}} that maps ${\displaystyle h}${\displaystyle h} to ${\displaystyle k}${\displaystyle k} lies in ${\displaystyle I}${\displaystyle I}. Define ${\displaystyle S_{h,f}}${\displaystyle S_{h,f}} to be the rank-1 operator that maps ${\displaystyle h}${\displaystyle h} to ${\displaystyle f}${\displaystyle f}, and ${\displaystyle S_{g,k}}${\displaystyle S_{g,k}} analogously. Then

${\displaystyle S_{h,k}=S_{g,k}TS_{h,f},,円}${\displaystyle S_{h,k}=S_{g,k}TS_{h,f},,円}

which means ${\displaystyle S_{h,k}}${\displaystyle S_{h,k}} is in ${\displaystyle I}${\displaystyle I} and this verifies the claim.

Some examples of two-sided *-ideals in ${\displaystyle L(H)}${\displaystyle L(H)} are the trace-class, Hilbert–Schmidt operators, and compact operators. ${\displaystyle F(H)}${\displaystyle F(H)} is dense in all three of these ideals, in their respective norms.

Since any two-sided ideal in ${\displaystyle L(H)}${\displaystyle L(H)} must contain ${\displaystyle F(H)}${\displaystyle F(H)}, the algebra ${\displaystyle L(H)}${\displaystyle L(H)} is simple if and only if it is finite dimensional.

A finite-rank operator ${\displaystyle T:U\to V}${\displaystyle 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

${\displaystyle Th=\sum _{i=1}^{n}\langle u_{i},h\rangle v_{i}\quad {\mbox{for all}}\quad h\in U,}${\displaystyle Th=\sum _{i=1}^{n}\langle u_{i},h\rangle v_{i}\quad {\mbox{for all}}\quad h\in U,}

where now ${\displaystyle v_{i}\in V}${\displaystyle v_{i}\in V}, and ${\displaystyle u_{i}\in U'}${\displaystyle u_{i}\in U'} are bounded linear functionals on the space ${\displaystyle U}${\displaystyle U}.

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

• Operator theory

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