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An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c0235101.png" /> defined on a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c0235102.png" /> of a topological vector space with values in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c0235103.png" />, such that every bounded subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c0235104.png" /> is mapped by it into a pre-compact set (cf. [[Pre-compact space|Pre-compact space]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c0235105.png" />. If, in addition, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c0235106.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c0235107.png" />, then it is called completely continuous on this set. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c0235108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c0235109.png" /> are Banach or, more generally, bornological spaces and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351010.png" /> is linear, the concept of a compact and a completely-continuous operator are the same. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351011.png" /> is a compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351012.png" /> is a continuous operator, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351014.png" /> are compact operators, so that the set of compact operators is a two-sided ideal in the ring of all continuous operators. In particular, a compact operator does not have a continuous inverse. The property of compactness plays an essential role in the theory of fixed points of an operator and in the study of its spectrum, which in this case has a number of "good"  properties.
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An operator $ A $ defined on a subset $ M $ of a topological vector space $ X $, with values in a topological vector space $ Y $, such that every bounded subset of $ M $ is mapped by it into a [[Pre-compact space|pre-compact]] subset of $ Y $. If, in addition, the operator $ A $ is continuous on $ M $, then it is called '''completely continuous''' on this set. In the case when $ X $ and $ Y $ are Banach or, more generally, bornological spaces and the operator $ A: X \to Y $ is linear, the concepts of a compact operator and of a completely-continuous operator are the same. If $ A $ is a compact operator and $ B $ is a continuous operator, then $ A \circ B $ and $ B \circ A $ are compact operators, so that the set of compact operators is a two-sided ideal in the ring of all continuous operators. In particular, a compact operator does not have a continuous inverse. The property of compactness plays an essential role in the theory of fixed points of an operator and in the study of its spectrum, which, in this case, has a number of ‘good’ properties.
 
 
Examples of compact operators are the Fredholm integral operators (cf. [[Integral operator|Integral operator]])
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351015.png" /></td> </tr></table>
 
  
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Examples of compact operators are the Fredholm [[Integral operator|integral operator]]
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$$
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A(x) = \int_{a}^{b} K(t,s) ~ x(s) ~ \mathrm{d}{s};
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$$
 
the Hammerstein operator
 
the Hammerstein operator
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351016.png" /></td> </tr></table>
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A(x) = \int_{a}^{b} K(t,s) ~ g(s,x(s)) ~ \mathrm{d}{s};
 
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$$
 
and the Urysohn (Uryson) operator
 
and the Urysohn (Uryson) operator
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$$
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A(x) = \int_{a}^{b} K(t,s,x(s)) ~ \mathrm{d}{s},
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$$
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in certain function spaces, under suitable restrictions on the functions $ K(t,s) $, $ g(t,u) $ and $ K(t,s,u) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351017.png" /></td> </tr></table>
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====References====
  
in certain function spaces, under suitable restrictions on the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023510/c02351020.png" />.
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<table>
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<TR><TD valign="top">[1]</TD><TD valign="top"> L.A. Lyusternik, V.I. Sobolev, “Elements of functional analysis”, Hindushtan Publ. Comp. (1974). (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD><TD valign="top"> K. Yosida, “Functional analysis”, Springer (1980), Chapt. 8, §1.</TD></TR>
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<TR><TD valign="top">[3]</TD><TD valign="top"> W. Rudin, “Functional analysis”, McGraw-Hill (1973).</TD></TR>
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<TR><TD valign="top">[4]</TD><TD valign="top"> M.A. Krasnosel’skii et al., “Integral operators and spaces of summable functions”, Noordhoff (1976). (Translated from Russian)</TD></TR>
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</table>
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "Elements of functional analysis" , Hindushtan Publ. Comp.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Rudin,  "Functional analysis" , McGraw-Hill  (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.A. Krasnosel'skii,  et al.,  "Integral operators and spaces of summable functions" , Noordhoff  (1976)  (Translated from Russian)</TD></TR></table>
 
 
  
 
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<table>
====Comments====
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<TR><TD valign="top">[a1]</TD><TD valign="top"> A.E. Taylor, D.C. Lay, “Introduction to functional analysis”, Wiley (1980).</TD></TR>
 
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<TR><TD valign="top">[a2]</TD><TD valign="top"> N. Dunford, J.T. Schwartz, “Linear operators. General theory”, '''1''', Interscience (1958).</TD></TR>
 
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</table>
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.E. Taylor,   D.C. Lay,   "Introduction to functional analysis" , Wiley (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience (1958)</TD></TR></table>
 

Latest revision as of 05:44, 7 December 2016

An operator $ A $ defined on a subset $ M $ of a topological vector space $ X $, with values in a topological vector space $ Y $, such that every bounded subset of $ M $ is mapped by it into a pre-compact subset of $ Y $. If, in addition, the operator $ A $ is continuous on $ M $, then it is called completely continuous on this set. In the case when $ X $ and $ Y $ are Banach or, more generally, bornological spaces and the operator $ A: X \to Y $ is linear, the concepts of a compact operator and of a completely-continuous operator are the same. If $ A $ is a compact operator and $ B $ is a continuous operator, then $ A \circ B $ and $ B \circ A $ are compact operators, so that the set of compact operators is a two-sided ideal in the ring of all continuous operators. In particular, a compact operator does not have a continuous inverse. The property of compactness plays an essential role in the theory of fixed points of an operator and in the study of its spectrum, which, in this case, has a number of ‘good’ properties.

Examples of compact operators are the Fredholm integral operator $$ A(x) = \int_{a}^{b} K(t,s) ~ x(s) ~ \mathrm{d}{s}; $$ the Hammerstein operator $$ A(x) = \int_{a}^{b} K(t,s) ~ g(s,x(s)) ~ \mathrm{d}{s}; $$ and the Urysohn (Uryson) operator $$ A(x) = \int_{a}^{b} K(t,s,x(s)) ~ \mathrm{d}{s}, $$ in certain function spaces, under suitable restrictions on the functions $ K(t,s) $, $ g(t,u) $ and $ K(t,s,u) $.

References

[1] L.A. Lyusternik, V.I. Sobolev, “Elements of functional analysis”, Hindushtan Publ. Comp. (1974). (Translated from Russian)
[2] K. Yosida, “Functional analysis”, Springer (1980), Chapt. 8, §1.
[3] W. Rudin, “Functional analysis”, McGraw-Hill (1973).
[4] M.A. Krasnosel’skii et al., “Integral operators and spaces of summable functions”, Noordhoff (1976). (Translated from Russian)

References

[a1] A.E. Taylor, D.C. Lay, “Introduction to functional analysis”, Wiley (1980).
[a2] N. Dunford, J.T. Schwartz, “Linear operators. General theory”, 1, Interscience (1958).
How to Cite This Entry:
Compact operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Compact_operator&oldid=13410
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article