APPROXIMATION OF NARROW OPERATORS ON THE SPACE L1 BY RANK ONE OPERATORS

Abstract

The note is devoted to the study of further properties of narrow operators, defined on the Lebesgue space L₁ and acting to an arbitrary Banach space. We investigate a new property, which asserts that any narrow operator on an ``essential’’ part of the domain space is arbitrary close in the sense of operator norm, to a rank one operator. ``Essential’’ means a subspace, which is isometrically isomorphic to L₁ itself and on which the operator ``almost’’ attains its norm. It looks somewhat strange, because there are isomorphic embeddings among narrow operators, and every linear bounded operator on the space L<sub>p</sub> with p>1 can be represented as a sum of two narrow operators. The proof of the main result uses Rosenthal’s lemma on the set of vectors on the space L1, at which the operator ``almost’’ attains its norm, and the Schauder bases technique. Finally we provide some examples of narrow operators with their corresponding approximations. Among these examples there is a strictly narrow operator, namely, the operator of conditional mathematical expectation with respect to the sub-ϭ-algebra of sets X˟[0,1], where X is an arbitrary Borel subset of [0,1], which is approximated by means of its restriction. There is an implicitly formulated open problem of constructive approximation of a certain compact operator with infinite-dimensional images, which is represented as a power series by the powers of two and the Rademacher system on the interval [0,1], for which the evaluation of the norm is a computationally difficult problem.