Transitive systems of linear operators on a Banach space

Abstract

Let X be an infinite-dimensional Banach space. Mackey2 has shown, under more general conditions, that if xl, , * * xn, yl, * , Yn are any two sets of n linearly independent elements in X, then there exists an isomorphism T of X with itself such that T(xi) = yi, i =1, , n. In other words the collection of isomorphisms is transitive for linearly independent sets of elements of ?&. This property is shared by many other sets of linear operators, for example by the set of all operators with finite-dimensional range in X. These two sets of linear operators are semi-groups.3 In this note we give a condition for the transitivity property above which is necessary for all sets of linear operators and which while not in general sufficient is so for semi-groups. In this result the operators need not be assumed to be bounded or defined everywhere in X. C(l) will be used to designate the collection of all bounded linear operators with domain X and range in 3X.