Hille-Tamarkin operator

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