Note on $F$-operators in locally convex spaces

The theory of F-operators in Banach spaces has been developed by several authors (cf. the references in [ΊΓ], IΓC)According to [5J, SL closed, normally solvable, linear mapping with finite ^-characteristic is called an F-operator. It is the purpose of this paper to generalize this notion of F-operator to locally convex spaces so that we may maintain a number of the basic results known in the case of Banach spaces. For the continuous F-operators, such an attempt has been made by H. Schaefer [9] and then by A. Deprit [4]. Our main concern here is the discussion of a general theory of F-operators: characterization of F-operators, the index theorem for a product, and so on.