Operators dense embedding and quasicomplements of subspaces of Banach spaces
Let \(E\) be a separable Banach space and \(E_ 1\) its infinite-dimensional and infinite-codimensional closed subspace. A quasicomplement \(E_ 2\) to \(E_ 1\) is called compactly adjacent to the subspace \(E_ 1\subset E\) if the restriction \(\varphi| E_ 2\) of the quotient map \(\varphi: E\to E/E_ 1\) on the subspace \(E_ 2\) is a compact dense embedding from \(E_ 2\) to \(E/E_ 1\). It is proved that if \(E_ 2\) is a quasicomplement to \(E_ 1\) then there exists a subspace \(E_ 3\subset E_ 2\) such that \(E_ 3\) is a compactly adjacent quasicomplement to \(E_ 1\). The proof is based on the following theorem (it generalizes a known theorem of Kato). Theorem: Let \(T: E\to X\) be a linear bounded operator such that the restrictions of \(T\) to any subspace of \(E\) of finite codimension is not an isomorphism. Then for every \(\varepsilon> 0\) there is a subspace \(N\subset E\) with a basis so that (1) \(T| N\) is compact, (2) \(\text{cl} T(N)= \text{cl} T(E)\), (3) \(\| T| N\|<\varepsilon\). Various properties of restrictions of a dense embedding operator \(T: E\to X\) (it means that \(\text{Ker }T= 0\) and \(R(T)\) is a dense subspace in \(X\)) on subspaces of \(E\) are described. For example, it is shown that if \(E= H\) is a Hilbert space then there exists a decomposition of \(H\) into an orthogonal sum of two subspaces \(H= H_ 1\oplus H_ 2\) such that both restrictions \(T| H_ 1\) and \(T| H_ 2\) are operators dense embedding from \(H_ 1\) and \(H_ 2\) to \(X\).
restrictions of a dense embedding operator
scientific article; zbMATH DE number 687072