Bukhvalov theorem

on kernel operators

The theory of kernel operators (cf. also Kernel of an integral operator) was essentially influenced by N. Dunford and B.J. Pettis [a2] around 1940. Other important results were obtained at about the same time by L.V. Kantorovich and B.Z. Vulikh. Among other results, it was shown that every bounded linear operator $T : L ^ { 1 } ( \mu ) \rightarrow L ^ { p } ( \nu )$, $1 < p \leq \infty$, is a kernel operator in the sense that there exists a measurable function $K$ such that $K ( ., s ) \in L ^ { 1 } ( \mu )$ for almost all $s$ and $(\; f \mapsto \int K ( t , \cdot ) f ( t ) d \mu ( t ) = T f ) \in L ^ { p } ( \nu )$.

This result is known as Dunford's theorem. In the decades following these first results, kernel operators were intensively studied. While the first results were mainly concerned with a single kernel operator or were in the spirit of Dunford's theorem, the investigation of the structure of the space of all kernel operators in the space of all regular operators began in the 1960s, following the study of Banach function spaces (cf. also Banach function space), or even ideal spaces in the space of measurable functions.

Under very general assumptions it was shown that the kernel operators of $T : L \rightarrow M$ form a band in the space of all regular operators, where $L$ and $M$ are ideals in the space of measurable functions $M ( \mu )$ or $M ( \nu )$, respectively. At that time many properties of kernel operators were known. It was A.V. Bukhvalov who gave [a1] a simple characterization of kernel operators, as follows.

Let $( \Omega , A , \mu )$, $( \Omega _ { 1 } , A _ { 1 } , \nu )$ be measure spaces (cf. also Measure space), let $L \subset M ( \mu )$ and $M \subset M ( \nu )$ be ideals such that the support of the Köthe dual $L ^ { \times }$ is all of $\Omega$. Then for every linear operator $T : L \rightarrow M$ the following conditions are equivalent:

i) $T$ is a kernel operator.

ii) If $( f _ { n } ) _ { n = 1 } ^ { \infty } \subset L _ { + }$ is an order-bounded sequence which is star convergent, then the sequence $( T f _ { n } ) _ { n = 1 } ^ { \infty } \subset M$ is convergent almost everywhere. Here, a sequence $( f _ { n } ) _ { n = 1 } ^ { \infty } $ said to be star convergent to some $f$ if every subsequence of the sequence $( f _ { n } ) _ { n = 1 } ^ { \infty } $ contains a subsequence $( h _ { n } ) _ { n = 1 } ^ { \infty } 1$ such that $h _ { n} \rightarrow f$ almost everywhere as $n \rightarrow \infty$. Consequently, $f _ { n } \rightarrow ^ { * } f$, as $n \rightarrow \infty$, if and only if $f _ { n } \rightarrow f$ in the measure $\mu$ on every subset of finite measure.

While the proof of i)$\Rightarrow$ii) is a simple consequence of the Lebesgue convergence theorem (cf. also Lebesgue theorem), the proof of ii)$\Rightarrow$i) requires many results concerning the structure of the space of kernel operators in the space of all regular operators. A simplified version of the proof is due to A.R. Schep [a4].

Bukhvalov's theorem is a powerful tool in the study of operators between Banach function spaces. In particular, his characterization of kernel operators leads to simple proofs of many classical results, such as Dunford's theorem and generalizations of it. For more information, see [a3], Sect. 3.3, or [a5].

References

[a1]	A.V. Bukhvalov, "Integral representations of linear operators" J. Soviet Math. , 8 (1978) pp. 129–137
[a2]	N. Dunford, J.B. Pettis, "Linear operators on summable functions" Trans. Amer. Math. Soc. , 47 (1940) pp. 323–392
[a3]	P. Meyer-Nieberg, "Banach lattices" , Springer (1991)
[a4]	A.R. Schep, "Kernel operators" PhD Thesis Univ. Leiden (1977)
[a5]	A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983)