Difference between revisions of "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==== | ====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 |
Hille-Tamarkin operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hille-Tamarkin_operator&oldid=22579