Let G be a locally compact group with left Haar measure μ, and let L(G) be the convolution Banach algebra of integrable functions on G with respect to μ. In this paper we are concerned with the investigation of the structure of G in terms of analytic semigroups in L(G).

We survey some recent results on functional calculus for generators of holomorphic semigroups, which have been obtained using versions of fractional derivation of Riemann-Liouville or Weyl type. Such a calculus allows us to give tight estimates even in concrete L¹ examples.

We introduce a notion of analytic generator for groups of unbounded operators, on Banach modules, arising from Esterle’s quasimultiplier theory. Characterizations of analytic generators are given in terms of the existence of certain functional calculi. This extends recent results about C₀ groups of bounded operators. The theory is applicable to sectorial operators, representations of ${H}^{\infty}$, and integrated groups.

We study the relation between standard ideals of the convolution Sobolev algebra ${\u208a}^{\left(n\right)}\left(t\u207f\right)$ and the convolution Beurling algebra L¹((1+t)ⁿ) on the half-line (0,∞). In particular it is proved that all closed ideals in ${\u208a}^{\left(n\right)}\left(t\u207f\right)$ with compact and countable hull are standard.

We establish an inversion formula of Post-Widder type for ${\lambda}^{\alpha}$-multiplied vector-valued Laplace transforms (α > 0). This result implies an inversion theorem for resolvents of generators of α-times integrated families (semigroups and cosine functions) which, in particular, provides a unified proof of previously known inversion formulae for α-times integrated semigroups.

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