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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h1101901.png" /> be an [[Integral operator|integral operator]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h1101902.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h1101903.png" />, i.e., there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h1101904.png" />-measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h1101905.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h1101906.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h1101907.png" /> a.e. on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h1101908.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h1101909.png" /> is called a Hille–Tamarkin operator if
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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/h/h110/h110190/h11019010.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h11019011.png" />. By taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h11019012.png" /> one obtains the class of Hilbert–Schmidt operators (cf. [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]]). Replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h11019013.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h11019014.png" />, by arbitrary Banach function norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h11019015.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110190/h11019016.png" />, one obtains the class of Hille–Tamarkin operators between Banach function spaces (sometimes called integral operators of finite double norm, see [[#References|[a1]]]). Under some mild hypotheses on the norms, one can show that Hille–Tamarkin operators have rather strong compactness properties (see [[#References|[a2]]]).
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Let  $  T $
 +
be an [[Integral operator|integral operator]] from  $  L _ {p} ( Y, \nu ) $
 +
into  $  L _ {q} ( X, \mu ) $,
 +
i.e., there exists a  $  ( \mu \times \nu ) $-
 +
measurable function  $  T ( x,y ) $
 +
on  $  X \times Y $
 +
such that  $  Tf ( x ) = \int {T ( x,y ) }  {d \nu ( y ) } $
 +
a.e. on  $  X $.  
 +
Then  $  T $
 +
is called a Hille–Tamarkin operator if
 +
 
 +
$$
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\int\limits {\left ( \int\limits {\left | {T ( x,y ) } \right | ^ {p  ^  \prime  } }  {d \nu ( y ) } \right ) ^ { {q / {p  ^  \prime  } } } }  {d \mu ( x ) } < \infty,
 +
$$
 +
 
 +
where  $  {1 / p } + {1 / { {p  ^  \prime  } } } = 1 $.  
 +
By taking $  p = q = 2 $
 +
one obtains the class of Hilbert–Schmidt operators (cf. [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]]). Replacing $  \| \cdot \| _ {p} $,  
 +
respectively $  \| \cdot \| _ {q} $,  
 +
by arbitrary Banach function norms $  \rho _ {2} $,  
 +
respectively $  \rho _ {1} $,  
 +
one obtains the class of Hille–Tamarkin operators between Banach function spaces (sometimes called integral operators of finite double norm, see [[#References|[a1]]]). Under some mild hypotheses on the norms, one can show that Hille–Tamarkin operators have rather strong compactness properties (see [[#References|[a2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.C. Zaanen,  "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.R. Schep,  "Compactness properties of Carleman and Hille–Tamarkin operators"  ''Canad. J. Math.'' , '''37'''  (1985)  pp. 921–933</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.C. Zaanen,  "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.R. Schep,  "Compactness properties of Carleman and Hille–Tamarkin operators"  ''Canad. J. Math.'' , '''37'''  (1985)  pp. 921–933</TD></TR></table>

Latest revision as of 22:10, 5 June 2020


Let $ T $ be an integral operator from $ L _ {p} ( Y, \nu ) $ into $ L _ {q} ( X, \mu ) $, i.e., there exists a $ ( \mu \times \nu ) $- measurable function $ T ( x,y ) $ on $ X \times Y $ such that $ Tf ( x ) = \int {T ( x,y ) } {d \nu ( y ) } $ a.e. on $ X $. Then $ T $ is called a Hille–Tamarkin operator if

$$ \int\limits {\left ( \int\limits {\left | {T ( x,y ) } \right | ^ {p ^ \prime } } {d \nu ( y ) } \right ) ^ { {q / {p ^ \prime } } } } {d \mu ( x ) } < \infty, $$

where $ {1 / p } + {1 / { {p ^ \prime } } } = 1 $. By taking $ p = q = 2 $ one obtains the class of Hilbert–Schmidt operators (cf. Hilbert–Schmidt operator). Replacing $ \| \cdot \| _ {p} $, respectively $ \| \cdot \| _ {q} $, by arbitrary Banach function norms $ \rho _ {2} $, respectively $ \rho _ {1} $, one obtains the class of Hille–Tamarkin operators between Banach function spaces (sometimes called integral operators of finite double norm, see [a1]). Under some mild hypotheses on the norms, one can show that Hille–Tamarkin operators have rather strong compactness properties (see [a2]).

References

[a1] A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983)
[a2] A.R. Schep, "Compactness properties of Carleman and Hille–Tamarkin operators" Canad. J. Math. , 37 (1985) pp. 921–933
How to Cite This Entry:
Hille-Tamarkin operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hille-Tamarkin_operator&oldid=22579
This article was adapted from an original article by A.R. Schep (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article