Well-bounded operators on general Banach spaces

The theory of well-bounded operators has found many applications and formed deep connections with other areas of mathematics. For example, it has been applied successfully to Sturm-Liouville theory, Fourier analysis and multiplier theory (see [2] and [4]). Although the theory of well-bounded operators is well established, there are a number of unresolved and interesting questions, which are potentially fruitful areas for further research; there are also a few errors in the literature. The general aims of this work are to answer some of these questions, to correct and clarify certain aspects of the theory, and to establish a more complete well-bounded operator theory including a dual theory on general Banach spaces and a theory of compact well-bounded operators. In particular, we show that on any Banach space X, every well-bounded operator which is decomposable in X is of type (A), and that on a very large class of nonreflexive spaces, there exists a well-bounded operator which is not of type (B). We also discuss the properties of well-bounded operators on some special class of Banach spaces