On quotients and restrictions of generalized scalar operators
The authors show that an operator \(T\in L(X)\) on a Banach space X is a quotient of a generalized scalar operator if and only if \[ (*)\quad T_ z: {\mathcal E}'({\mathbb{C}}){\tilde \otimes}X\to {\mathcal E}'({\mathbb{C}}){\tilde \otimes}X,\quad u\mapsto (z-T)u \] is surjective. If \({\mathbb{C}}\) is replaced by the real numbers \({\mathbb{R}}^ a \)spectral characterization of those T is given obeying this modified condition (*). This characterization is used to solve division problems for vector valued distributions on \({\mathbb{R}}\). Finally, an example is exhibited of an operator, which is both a restriction and a quotient of a generalized scalar operator, but which is not even strongly decomposable. This is in strict contrast to the situation with decomposable operators.
quotient of a generalized scalar operator
division problems for vector valued distributions
an operator, which is both a restriction and a quotient of a generalized scalar operator, but which is not even strongly decomposable
scientific article; zbMATH DE number 4111384
scientific article