A characterization of Enflo-operators. (English) Zbl 0549.47015
Operator algebras, ideals, and their applications in theoretical physics, Proc. int. Conf., Leipzig 1983, Teubner-Texte Math. 67, 220-231 (1984).
[For the entire collection see Zbl 0535.00014.]
A bounded lienear operator \(T:L_ 1(X,\mu)\to L_ 1(X,\mu)\) is called an Enflo-operator if there is a subspace M of \(L_ 1(X,\mu)\), isomorphic to \(L_ 1(X,\mu)\), such that \(T|_ M\) is an isomorphism into. In this note we show that T is not an Enflo-operator if and only if every non-atomic sublattice of \(L_ 1(X,\mu)\) contains a non-atomic sublattice F such that \(T|_ F\) is compact and we give a second characterization in terms of an equi-integrability condition modeled after the well known characterization of weakly compact subsets of \(L_ 1(X,\mu)\).
A bounded lienear operator \(T:L_ 1(X,\mu)\to L_ 1(X,\mu)\) is called an Enflo-operator if there is a subspace M of \(L_ 1(X,\mu)\), isomorphic to \(L_ 1(X,\mu)\), such that \(T|_ M\) is an isomorphism into. In this note we show that T is not an Enflo-operator if and only if every non-atomic sublattice of \(L_ 1(X,\mu)\) contains a non-atomic sublattice F such that \(T|_ F\) is compact and we give a second characterization in terms of an equi-integrability condition modeled after the well known characterization of weakly compact subsets of \(L_ 1(X,\mu)\).
MSC:
| 47B38 | Linear operators on function spaces (general) |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
