Regularity of Banach algebras generated by analytic semigroups satisfying some growth conditions

Abstract:

We show that if a commutative complex Banach algebra $A$ is generated by a nonzero analytic semigroup $({a^t})\operatorname {Re} t > 0$ satisfying \[ \int {\begin {array}{*{20}{c}} { + \infty } \\ { - \infty } \\ \end {array} } \frac {{{{\log }^ + }\left \| {{a^{1 + it}}} \right \|}}{{1 + {t^2}}}dt < + \infty ,\], then $A$ is regular in Shilov’s sense.