SOME PROPERTIES OF \(\mu\)-APPROXIMATE \(l_1\) SEQUENCES IN BANACH SPACES

Abstract

The main result in this paper is the following: If \(X\) is a Banach space, which contains an \(\mu\)-approximative \(l_1\) system of vectors then there exists a subspace \(L\) and a weighted shift operator \(T : L \to L\) with a weighted sequence \((\lambda_i)\) such that \(\inf_i(1-\varepsilon_i)\cdot|\lambda_i|\|x\|-K \leq \|Tx\| \leq \sup_i|\lambda_i|\|x\|\), \(K\)-constant, for every \(x\) in \(L\) and for every decreasing sequence \(0<(\varepsilon_i)<1\) of real numbers, in case where \(X=L_2[R]\) the norm of operator is estimate in whole space \(X=L_2\).