View source for Hille-Tamarkin operator

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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

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

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====
<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>