View source for Compact operator

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.

Examples of compact operators are the Fredholm integral operators (cf. [[Integral operator|Integral operator]])

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the Hammerstein operator

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and the Urysohn (Uryson) operator

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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" />.

====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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====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>