Rotation operators with multiplication and some of their generalizations

Abstract

We start with an example, clarifying the topic of this note. In the Hardy space HP(D), i < p < ~, we consider an operator T of the form (Tx)(z)= a~)x(e~Oz), z ~ D, where 8 is incommensurable with ~, while a is an element of the disk-algebra A. The spectrum of the operator T can be easily computed [i] and it turns out that in the case when the multiplier a is invertible we have a(T)= a(0)T (T is the unit circumference), while for a noninvertible a we have o(T) = ae(0)D (a e is the exterior part of a). In a similar manner one computes the spectrum of the rotation operator with multiplication and in many other spaces of functions, analytic in the circle (see, e.g., [2]), which suggests that the corresponding fact must be true in a sufficiently general situation. This note contains an abstract scheme connected with one of the possible generalizations and we give applications to operators in spaces of analytic functions of one and several variables.