Tauberian convolution operators acting on L1(G)

Convolution operators on group algebras which are tauberian or cotauberian

“…Indeed, it was proved in [3] that there exists an atomic measure μ 0 ∈ M (T) such that T μ 0 is an injective non-tauberian operator, where T denotes the unit circle. It is enough to choose μ 0 such that its Fourier-Stieltjes transform μ 0 satisfies 0 ∈ μ 0 (Z)\ μ 0 (Z).…”

“…with non-closed range) tauberian operators T : L 1 (μ) → L 1 (μ) have been found in [13], answering a question in [9]. In [3] we considered the case in which T is a convolution operator T μ acting on the group algebra L 1 (G), where G be a locally compact abelian group. We proved that T μ tauberian implies T μ invertible when G is non-compact, and that T μ tauberian implies T μ Fredholm when G is compact and the singular continuous part μ sc of μ with respect to the Haar measure on G is zero.…”

“…In [3], our main tool was a result in [8] characterizing the tauberian operators T : L 1 (μ) → Y in terms of the action of T over sequences of normalized disjoint functions in L 1 (μ). Here we follow a more algebraic approach, based in the characterization of the convolution operators T μ as the multipliers of the group algebra L 1 (G) [15,Chapter 0].…”

“…We show first that T μ cotauberian implies T μ tauberian. In the case G non-compact, T μ tauberian implies T μ invertible by a result in [3]. In the case G compact, L 1 (G) * * is a Banach algebra in which L 1 (G) is a closed ideal, hence L 1 (G) * * /L 1 (G) is a Banach algebra, and we prove that T μ cotauberian implies that the operator induced by T * * μ on the quotient algebra L 1 (G) * * /L 1 (G) is bijective.…”