[Author]
 O. I. Reinov

[Title]
 Notes on tensor products of operators in Lebesgue spaces

[AMS Subj-class]
 47B10 Hilbert--Schmidt operators, trace class operators,
       nuclear operators, p-summing operators, etc.

[Abstract]
   We investigate a question connected with computing the norms of
 tensor product of operators acting between Lebesgue function spaces:
 is it true that, if we have two operators $A: L^p(\mu)\to L^q(\nu)$
 and $B: L^p(\mu)\to L^q(\nu)$, then the norm of their tensor product
 $A\otimes B: L^p(\mu\otimes\mu)\to L^q(\nu\otimes\nu)$ coincides
 with the product $\|A\|\,\|B\|$ of their norms.
   We formulate results which show that the answer is positive if and
 only if $1\le p\le q\le+\infty$. The results are answering a question
 posed to the author by Professor Ja.Yu.Nikitin in the end of May, 2000,
 and can be applied, in particular, to solve some problems in the theory
 of probability.
   The positive part of our results can be proved by using abstract
 tensor product techniques in normed spaces, but also there is
 a surprising simple elementary proof using only some facts from the
 classical Lebesgue measure theory. As to negative part of our results
 ("counterexamples"), to obtain them we use some estimates in the theory
 of finite rank $p$-summing operators in normed spaces. The last proof
 consists of eleven Lemmas, in which we consider, step by step,
 different possible cases such as the cases when $1<q<p<2$,
 $1<p<2<q<+\infty$ etc.

[Comments]
 AmSTeX, 27 pp., Russian

[Contact e-mail]
 orein@orein.usr.pu.ru