LINEAR OPERATORS WITH EMPTY SPECTRUM

Abstract

In this note we are interested in showing a simple property which characterizes linear operators having empty spectrum on complex Banach spaces. Although it is rather well known that such operators exist, the current literature deals with very few such examples,only as an exercise topic. Our characterization says that such operators are in one-to-one correspondence with injective quasinilpotent operators. This means that the purely unbounded property of linear operators can be reduced to the bounded one. In fact, we obtain a simple way to test and construct examples of such operators and then proceed to make observations of some interest from the function-theoretic point of view. For instance, we define an analogue of the order in the theory of entire functions and show the existence of operators of all order types.