Majorization on \(\ell^\infty\) and on its closed linear subspace \(\mathfrak c\), and their linear preservers (Q448385)

The theory of majorizations on different vector spaces is an interesting and intensively developing topic during the years. The infinite-dimensional version of this theory is also important.NEWLINENEWLINEIn the paper under review the authors extend the definition of majorization to the Banach space of all bounded real sequences in \(l^\infty\) using the notion of a doubly-stochastic operators on \(l^\infty\).NEWLINENEWLINEThe properties of the introduced majorization are investigated on the space \(l^\infty\) and its closed linear subspaces including the space of all convergent sequences and the space \(l_0\) of all sequences convergent to zero.NEWLINENEWLINEIn addition the structure of linear preservers of majorization on the Banach spaces \(c\) and \(c_0\) is obtained. It is proved that the structure of these maps on \(c_0\) is exactly the same as that on \(l^p\)-spaces with \(1<p<\infty\) with the only exception that the columns of the corresponding infinite matrix belong to \(c_0\) rather than to \(l^p\).NEWLINENEWLINEThe paper contains many interesting results and examples concerning linear preservers for the majorization on the space \(l^\infty\).