SOME PROPERTIES OF \(μ\)-APPROXIMATE \(l_1\) SEQUENCES IN BANACH SPACES - Lohaj, M; Braha, N. (Matematichki Bilten, 2003)
The main result in this paper is the following: If \(X\) is a Banach space, which contains an \(μ\)-approximative \(l_1\) system of vectors then there exists a subspace \(L\) and a weighted shift operator \(T : L → L\) with a weighted sequence \((λ_i)\) such that \(_i(1-ε_i)·|λ_i|\|x\|-K ≤ \|Tx\| ≤ _i|λ_i|\|x\|\), \(K\)-constant, for every \(x\) in \(L\) and for every decreasing sequence \(0<(ε_i)<1\) of real numbers, in case where \(X=L_2[R]\) the norm of operator is estimate in whole space \(X=L_2\).