Invariant subspaces of abstract multiplication operators

INVARIANT SUBSPACES OF ABSTRACT MULTIPLICATION OPERATORS by Hermann Flaschka We describe a class of operators on a Banach space ft whose members behave, in a sense, like multiplication operators, and consequently leave invariant a proper closed subspace of IB. One of the sufficient conditions for an operator to be such an "abstract multiplication" bears a striking resemblence to an assumption made by j. Wermer, who approached the invariantsubspace problem from a very different point of view. INVARIANT SUBSPACES OF ABSTRACT MULTIPLICATION OPERATORS by Hermann Flaschka §1. We want to describe a class of operators on a Banach space IB whose members behave, in a sense, like multiplication operators, and consequently leave invariant a proper closed subspace of ft -that is, they are intransitive. One of the sufficient conditions for an operator to be such an "abstract multiplication" bears a striking resemblance to an assumption made by J. Wermer [1], who approached the invariant-subspace problem from a very different (and rather more sophisticated) point of view. Our comments are presented in the hope that this connection, as well as the more general pattern which appears to emerge, may be more than superficial. Wermer considered operators whose deviation from being an isometry is limited. More precisely, he assumed that T and T~ are both bounded, and that either (A) ||T | | = 0 ( e ( n ) , n = 0 , + l , 0 < a < 1, and the spectrum of T contains at least two points, or (B) ||T|| = 0(|n| ), for some fixed k < oo . Such a. T jus intransitive. If a = 0 in (A) , or k = 0 in (B) , IIT || < K for all n; the space may then be renormed to make T an isometry, and intransitivity follows from a theorem of Godement [2] . The crucial estimate is that in (A); it is a reformulation of the requirement that