In Matrix Theory, the Spectral Mapping Theorem states that for a matrix A and a polynomial p, {Eigenvalues of p(A)} = p({Eigenvalues of A}). Spectral Theory generalises eigenvalues, invertibility and the spectral mapping theorem to Banach algebras. Axiomatic Spectral Theory takes this a step further to pseudo-invertibility and considers the associated spectral mapping theorem which has applications in various areas such as differential equations. This book develops a new pointwise axiomatic theory based on Muller's Regularites, sharing and extending several properties. The spectral mapping theorem is proved and extended to pointwise theory. Other forms of axiomatic spectra are shown to form Total Pointwise Regularities. Many sets are shown to have good spectral properties and some new sets are shown to be Global Regularities. The derived approach of Berkani's regularity theory is developed and axiomatised into a restricted theory that will always lead to a global regularity. Also, the systematic approach to regularities of Mbekhta and Muller are examined and extended to general Banach algebras by means of annihilators and the socle.

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