Summary: The notion of spectrum of an operator is one of the central concepts of operator theory. It is closely connected with the existence of a functional calculus which provides important information about the structure of Banach space operators. The situation for commuting \(n\)-tuples of Banach space operators is much more complicated. There are many possible definitions of joint spectra. However, the joint spectrum introduced by J. L. Taylor has a distinguished property – there exists a functional calculus for functions analytic on a neighborhood of this spectrum. The present paper gives a survey of basic properties of the Taylor spectrum and Taylor functional calculus.
For the entire collection see [Zbl 1325.47001].