Weyl’s Theorems for Some Classes of Operators (original) (raw)
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Acta Mathematica Hungarica, 2010
We study weakly compact left and right multipliers on the Banach algebra L ∞ 0 (G) * of a locally compact group G. We prove that G is compact if and only if L ∞ 0 (G) * has either a non-zero weakly compact left multiplier or a certain weakly compact right multiplier on L ∞ 0 (G) *. We also give a description of weakly compact multipliers on L ∞ 0 (G) * in terms of weakly completely continuous elements of L ∞ 0 (G) *. Finally we show that G is finite if and only if there exists a multiplicative linear functional n on L ∞ 0 (G) such that n is a weakly completely continuous element of L ∞ 0 (G) * .
Compact Left Multipliers on Banach Algebras Related to Locally Compact Groups
Bulletin of the Australian Mathematical Society, 2009
We deal with the dual Banach algebras L ∞ 0 (G) * for a locally compact group G. We investigate compact left multipliers on L ∞ 0 (G) * , and prove that the existence of a compact left multiplier on L ∞ 0 (G) * is equivalent to compactness of G. We also describe some classes of left completely continuous elements in L ∞ 0 (G) * .
Multipliers of Group Algebras of Vector-Valued Functions
Proceedings of the American Mathematical Society, 1981
Let G be a locally compact abelian group and A" be a Banach space. Let Ll(G, X) be the Banach space of X-valued Bochner integrable functions on G. We prove that the space of bounded linear translation invariant operators of L'(G, X) can be identified with L(X, M(G, X)), the space of bounded linear operators from X into M(G, X) where M(G, X) is the space of Jf-valued regular, Borel measures of bounded variation on G. Furthermore, if A is a commutative semisimple Banach algebra with identity of norm 1 then L\G, A) is a Banach algebra and we prove that the space of multipliers of L\G, A) is isometrically isomorphic to M{G, A). It also follows that if dimension of A is greater than one then there exist translationinvariant operators of Ll(G, A) which are not multipliers of L1(G, A).
2010
We study multiplier algebras for a large class of Banach algebras which contains the group algebra L 1 (G), the Beurling algebras L 1 (G, ω), and the Fourier algebra A(G) of a locally compact group G. This study yields numerous new results and unifies some existing theorems on L 1 (G) and A(G) through an abstract Banach algebraic approach. Applications are obtained on representations of multipliers over locally compact quantum groups and on topological centre problems. In particular, five open problems in abstract harmonic analysis are solved.
Generalised Weyl's Theorem for A Class of Operators Satisfying A Norm Condition II
Mathematical Proceedings of the Royal Irish Academy, 2006
For a Banach space operator T ∈ B(X), it is proved that if either T is an algebraically, totally hereditarily normaloid operator and the Banach space X is separable, or T satisfies the property that its quasinilpotent part H 0 (T − λ) = (T − λ) −p (0) for all complex numbers λ and some integer p ≥ 1, then f (T) satisfies generalized Weyl's theorem for every non-constant function f that is analytic on an open neighborhood of σ(T).
Multipliers with Closed Range on Regular Commutative Banach Algebras
Proceedings of the American Mathematical Society, 1994
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.