We study properties under which the domain of a closed derivation of a generalized B-algebra remains invariant under analytic functional calculus. For a complete, generalized B-algebra with jointly continuous multiplication, two sufficient conditions are assumed: that the unit of belongs to the domain of the derivation, along with a condition related to the coincidence of the (Allan) spectra for every element . Certain results are derived concerning the spectra for a general element of the domain, in the realm of a domain which is advertibly complete or enjoys the Q-property. For a closed -derivation of a complete GB-algebra with jointly continuous multiplication such that and a normal element of the domain, for every analytic function on a neighborhood of the spectrum of . We also give an example of a closed derivation of a GB-algebra which does not contain the identity element. A condition for a closed derivation of a GB-algebra to be the generator of a one-parameter group of automorphisms of is provided along with a generalization of the Lumer–Phillips theorem for complete locally convex spaces.
Banach J. Math. Anal.
12(4):
873-908
(October 2018).
DOI: 10.1215/17358787-2017-0060