CHARACTERISATIONS OF OPERATORS OF LOWER SEMI-FREDHOLM TYPE IN NORMED LINEAR SPACES

Let X and Y be normed linear spaces. A linear operator T: D(T) ⊂ X → Y is called an F-operator if its adjoint T′: D(T) ⊂ Y′ → D(T)' is a φ+ -operator, i.e. has closed range and finite dimensional-kernel. Characterisations of an F_-operator T are obtained in the general case and in the case when T is closable. Unbounded strictly cosingular operators are defined and shown to belong to the class of F_ -admissible pertubations whenever Y is complete.