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