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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.
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330:{\displaystyle M=\alpha \cdot uv^{*},\quad {\mbox{where}}\quad \|u\|=\|v\|=1\quad {\mbox{and}}\quad \alpha \geq 0.}
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655:{\displaystyle Th=\sum _{i=1}^{n}\alpha _{i}\langle h,v_{i}\rangle u_{i}\quad {\mbox{for all}}\quad h\in H,}
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1808:{\displaystyle Th=\sum _{i=1}^{n}\langle u_{i},h\rangle v_{i}\quad {\mbox{for all}}\quad h\in U,}
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A bounded linear functional is a particular case of a finite-rank operator, namely of rank one.
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must contain the finite-rank operators. This is not hard to prove. Take a non-zero operator
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is finite dimensional. Just as in the
Hilbert space case, it can be written in the form
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992:. In fact it is the minimal element among such ideals, that is, any two-sided *-ideal
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is convergent; a property that automatically holds for all finite-rank operators.
464:{\displaystyle Th=\alpha \langle h,v\rangle u\quad {\mbox{for all}}\quad h\in H,}
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From linear algebra, we know that a rectangular matrix, with complex entries,
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is dense in all three of these ideals, in their respective norms.
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are orthonormal bases. Notice this is essentially a restatement of
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is now countably infinite and the sequence of positive numbers
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842:, and one has the canonical form for compact operators.
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46:. Unsourced material may be challenged and removed.
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340:Exactly the same argument shows that an operator
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1947:. New York: Springer-Verlag. pp. 267–268.
188:{\displaystyle M\in \mathbb {C} ^{n\times m}}
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1889:are bounded linear functionals on the space
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1430:{\displaystyle S_{h,k}=S_{g,k}TS_{h,f},\,}
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106:Learn how and when to remove this message
142:Finite-rank operators on a Hilbert space
1921:
1664:Finite-rank operators on a Banach space
1496:Some examples of two-sided *-ideals in
972:, the algebra of bounded operators on
509:Therefore, by induction, an operator
7:
1930:"Finite Rank Operator - an overview"
1286:to be the rank-1 operator that maps
894:The family of finite-rank operators
44:adding citations to reliable sources
1128:. It suffices to have that for any
14:
879:{\textstyle \sum _{i}\alpha _{i}}
20:
1945:A course in functional analysis
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788:{\displaystyle \{\alpha _{i}\}}
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31:needs additional citations for
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1:
1569:Since any two-sided ideal in
1493:and this verifies the claim.
122:, a branch of mathematics, a
943:form a two-sided *-ideal in
733:singular value decomposition
1882:{\displaystyle u_{i}\in U'}
735:. This can be said to be a
499:{\displaystyle \alpha ,u,v}
1999:
1844:{\displaystyle v_{i}\in V}
742:Generalizing slightly, if
739:of finite-rank operators.
1531:Hilbert–Schmidt operators
1121:{\displaystyle f,g\neq 0}
724:{\displaystyle \{v_{i}\}}
691:{\displaystyle \{u_{i}\}}
1943:Conway, John B. (1990).
1693:{\displaystyle T:U\to V}
1153:{\displaystyle h,k\in H}
474:where the conditions on
138:is finite-dimensional.
1668:A finite-rank operator
1466:{\displaystyle S_{h,k}}
1352:{\displaystyle S_{g,k}}
1279:{\displaystyle S_{h,f}}
1186:{\displaystyle S_{h,k}}
128:bounded linear operator
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1060:{\displaystyle T\in I}
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845:Compact operators are
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55:"Finite-rank operator"
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1649:{\displaystyle L(H)}
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1620:{\displaystyle F(H)}
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1591:{\displaystyle L(H)}
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1559:{\displaystyle F(H)}
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1518:{\displaystyle L(H)}
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1089:{\displaystyle Tf=g}
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1034:{\displaystyle L(H)}
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965:{\displaystyle L(H)}
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916:{\displaystyle F(H)}
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124:finite-rank operator
40:improve this article
923:on a Hilbert space
849:only if the series
360:on a Hilbert space
120:functional analysis
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1359:analogously. Then
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890:Algebraic property
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1954:978-0-387-97245-9
1902:{\displaystyle U}
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1535:compact operators
1486:{\displaystyle I}
1319:{\displaystyle f}
1299:{\displaystyle h}
1246:{\displaystyle I}
1226:{\displaystyle k}
1206:{\displaystyle h}
1005:{\displaystyle I}
985:{\displaystyle H}
936:{\displaystyle H}
856:
831:{\displaystyle T}
811:{\displaystyle 0}
755:{\displaystyle n}
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542:{\displaystyle n}
522:{\displaystyle T}
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393:{\displaystyle 1}
373:{\displaystyle H}
353:{\displaystyle T}
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228:{\displaystyle M}
208:{\displaystyle 1}
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1706:bounded operator
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840:compact operator
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147:A canonical form
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1983:Operator theory
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737:canonical form
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1598:must contain
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734:
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27:
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38:Please help
33:verification
30:
1527:trace-class
847:trace class
380:is of rank
1916:References
1818:where now
1193:that maps
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66:newspapers
1869:∈
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96:June 2021
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