163.47.13.81: /* Clipping algorithms */

2014-12-25T10:33:31Z

Clipping algorithms

en>BattyBot: changed {{Unreferenced}} to {{Refimprove}} & general fixes using AWB (7940)

2012-02-02T00:12:38Z

changed {{Unreferenced}} to {{Refimprove}} & general fixes using AWB (7940)

New page

{{Unreferenced|date=December 2006}}
In [[functional analysis]], a branch of mathematics, a '''finite-rank operator''' is a [[bounded linear operator]] between [[Banach space]]s whose [[Range (mathematics)|range]] is finite-dimensional.

==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'' ∈ '''C'''''n'' × ''m'' has rank 1 if and only if ''M'' 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 ''T'' on a Hilbert space ''H'' is rank 1 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 ''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

:<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 {''ui''} and {''vi''} 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 {''αi''} accumulate only at 0, ''T'' is then a [[compact operator on Hilbert space|compact operator]], and one has the canonical form for compact operators.

If the series ∑''i'' ''αi'' is convergent, ''T'' is a [[trace class]] operator.

===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'' ∈ ''I'', then ''Tf'' = ''g'' for some ''f, g'' ≠ 0. It suffices to have that for any ''h, k'' ∈ ''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

:<math>S_{h,k} = S_{g,k} T S_{h,f}, \,</math>

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 operator]]s, and [[compact operator]]s. ''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 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 (mathematics)|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 \alpha_i \langle h, v_i\rangle u_i \quad \mbox{for all} \quad h \in U ,</math>

where now <math>u_i\in V</math>, and <math>v_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.

{{DEFAULTSORT:Finite Rank Operator}}
[[Category:Operator theory]]

Clip coordinates - Revision history

163.47.13.81: /* Clipping algorithms */

en>BattyBot: changed {{Unreferenced}} to {{Refimprove}} & general fixes using AWB (7940)