A signi?cant sector of the development of spectral theory outside
the classical area of Hilbert space may be found amongst at
multipliers de?ned on a complex commutative Banach algebra A.
Although the general theory of multipliers for abstract Banach
algebras has been widely investigated by several authors, it is
surprising how rarely various aspects of the spectral theory, for
instance Fredholm theory and Riesz theory, of these important
classes of operators have been studied. This scarce consideration
is even more surprising when one observes that the various aspects
of spectral t- ory mentioned above are quite similar to those of a
normal operator de?ned on a complex Hilbert space. In the last ten
years the knowledge of the spectral properties of multip- ers of
Banach algebras has increased considerably, thanks to the
researches undertaken by many people working in local spectral
theory and Fredholm theory. This research activity recently
culminated with the publication of the book of Laursen and Neumann
, which collects almost every thing that is known about the
spectral theory of multipliers.
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