On quotients and restrictions of generalized scalar operators

Abstract

In a preceding paper the authors characterized the continuous linear Banach space operators, which are up to similarity a restriction of an operator generalized scalar in the sense of Colojoară and Foias, as those operators T ∈ L( X), for which T z: E ( C , X)→ E ( C , X), ƒ → (z − T)ƒ is a topological monomorphism. In the present paper it is shown that an operator T ∈ L( X) is a quotient of a generalized scalar operator if and only if T z: : E ′( C ) ⊗X→ E ′( C ) ⊗X, u → (z − T)u is onto. The situation which arises, if the complex plane is replaced by the real line, is clarified and applications to division problems for vector valued distributions on the real line are given. An example of an operator is given, which is both a restriction and a quotient of a generalized scalar operator, but which is even not strongly decomposable.