Axiomatic theory of spectrum III. — Semiregularities

The notion of regularity in a Banach algebra was introduced and studied in [KM] and [MM]. A non-empty subset R of a unital Banach algebra A is called a regularity if it satisfies the following two conditions: (i) if a ∈ A and n ∈ N, then a ∈ R ⇔ a ∈ R, (ii) if a, b, c, d are mutually commuting elements of A satisfying ac + bd = 1A then ab ∈ R ⇔ a, b ∈ R. The axioms of regularities are weak enough so that there are plenty of examples that appear naturally in Banach algebras and operator theory. On the other hand they are strong enough so that they have interesting consequences, especially the spectral mapping theorem for the corresponding spectrum σR(a) = {λ ∈ C : a− λ / ∈ R}. In fact the axioms (i) and (ii) of regularities can be divided in two halves, each of them implying a one-way spectral mapping theorem. The aim of this paper is to study systematically semiregularities defined in this way. There are many natural examples of such classes that satisfy only one half of the axioms of regularities. The corresponding spectra include the exponential spectrum, the Weyl spectrum, T -Weyl spectrum, Kato essential spectrum, various essential spectra etc. All Banach algebras considered in this paper are complex and unital.