Subspaces in a couple of Banach spaces (Q1111129)

Let \(E_ 0\) and \(E_ 1\) be a couple of normally imbedded Banach spaces \((E_ 1\subset E_ 0)\). A closed subspace \(N\) of \(E_ 1\) is said to be polyclosed if \(N\) is closed in \(E_ 0\). The author investigates properties of the polyclosedsubspaces. The main results are the following: (1) for any separable Banach space \(X\) there exists a Banach space \(Y\subset X\) such that \(Y=M+N\), where \(M,N\) are infinite-dimensional polyclosed subspaces and \(``+''\) denotes a direct sum; (2) if \(E_ 1\) is separable and \(N\) is an infinite-dimensional polyclosed subspace, then there exists a polyclosed quasicomplement \(M\) for \(N\) in \(E_ 1;\) (3) if \(E_ 1\) is a separable Banach space, then \(E_ 1\) has a closed infinite-dimensional subspace \(N\) such that dim \(E_ 1/N=\infty\) and the closure of \(N\) in \(E_ 0\) coincides with \(E_ 0.\) The notion of a maximal polyclosed subspace is introduced and its properties (for normally imbedded Hilbert spaces) are studied. Various examples of polyclosed subspaces are given.