Multipliers with closed range on regular commutative Banach algebras

Abstract

Abstract. Conditions equivalent with closure of the range of a multiplier T, defined on a commutative semisimple Banach algebra A , are studied. A main result is that if A is regular then T2A is closed if and only if T is a product of an idempotent and an invertible. This has as a consequence that if A is also Tauberian then a multiplier with closed range is injective if and only if it is surjective. Several aspects of Fredholm theory and Kato theory are covered. Applications to group algebras are included.

When is the Range of a Multiplier on a Banach Algebra Closed?

“…The abstract version of Beurling's problem, the above Problem (M), has been studied mostly by K. B. Laursen and his coauthors in [1], [13] and [15]. The reader can find in the book [15, of Laursen and Neumann most of the known results about this problem up to the year 2000.…”

Weyl’s Theorems for Some Classes of Operators

“…Other important examples of regular and Tauberian Banach algebras are L p (G), 1 ≤ p < ∞, G a compact abelian group, and C 0 (Ω), the Banach algebra of continuous complexvalued functions on a locally compact Haudorff space Ω which vanish at infinity, see Rudin [23] Proof. If A is regular and Tauberian then a multiplier T ∈ M (A) with closed range is injective if and only if is surjective, see [3,Corollary 4.4], so that σ a (T ) = σ(T ). From this it follows that π a 00 (T ) = π 00 (T ).…”

Essential spectra through local spectral theory

“…dim ker(T − λ) = codim(T − λ)A < ∞. Obviously σ sF (T ) ⊆ σ w (T ), and the example of the disc algebra A(D) shows that these two spectra may well be distinct: for (T f)(z) := zf (z), where f ∈ A(D) we have that σ(T ) = D, and the interior of D consists of points λ for which (T − λ) is Fredholm, but the index is −1, hence σ w (T ) = D while σ sF (T ) = T. On the other hand, with additional assumptions the two spectra do coincide; for instance, if A is a regular semisimple commutative and Tauberian Banach algebra, then σ sF (T ) = σ w (T ) [2,Theorem 4.5]. Moreover, with these assumptions on A, as well as the assumption that ∆(A) contains no isolated points then σ sF (T ) = σ w (T ) = σ(T ).…”

“…Moreover, with these assumptions on A, as well as the assumption that ∆(A) contains no isolated points then σ sF (T ) = σ w (T ) = σ(T ). The technical terms used in these last statements are defined, for instance, in [2].…”