On regularities and Fredholm theory

“…In the early 1980's Harte [10] introduced Fredholm, Weyl and Browder theory relative to a unital homomorphism T : A → B between general unital Banach algebras A and B. Several authors have continued this investigation (see [11], [12], [18], [19], [15], [17], [4] and [5]).…”

Section: Introductionmentioning

confidence: 99%

r-FREDHOLM THEORY IN BANACH ALGEBRAS

Benjamin¹,

Laustsen²,

Mouton³

2018

Glasgow Math. J.

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Harte (1982, Math. Z. 179, 431–436) initiated the study of Fredholm theory relative to a unital homomorphism T: A → B between unital Banach algebras A and B based on the following notions: an element a ∈ A is called Fredholm if 0 is not in the spectrum of Ta, while a is Weyl (Browder) if there exist (commuting) elements b and c in A with a = b + c such that 0 is not in the spectrum of b and c is in the null space of T. We introduce and investigate the concepts of r-Fredholm, r-Weyl and r-Browder elements, where 0 in these definitions is replaced by the spectral radii of a and b, respectively.

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“…The nonzero spectrum offers a definition of "rank": we set ( [11], [18]) (0.5) rank(a) = rank A (a) = sup x∈A #σ (xa) = sup x∈A #σ (ax) ≤ ∞ .…”

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confidence: 99%

“…We might remark on a curious converse ( [18], [6]) to (0.10): any ideal J ⊆ A is a subset of the socle. This exactly parallels Smyth's observation [22] about ideals of algebraic elements.…”

mentioning

confidence: 99%

Adjugates in Banach algebras

Harte

Hernández²

2005

Proc. Amer. Math. Soc.

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Abstract. A simple formula for the adjugate of a block triangle offers an alternative route to the determinant theory for Banach algebras.Suppose A is a complex linear algebra, with identity 1 and invertible group A −1 . The radical of A,coincides with the intersection of the maximal left ideals, and also of the maximal right ideals. Whenthe algebra A is described as semisimple. The spectrum of an element a ∈ A is given bywe shall also writefor the nonzero spectrum. The nonzero spectrum offers a definition of "rank": we set ( Thus, if rank(a) ≤ 1, then there is a bounded linear functional τ a ∈ A * for whichIt is clear that if a = 0, then τ a is uniquely determined; obviously we take τ 0 = 0. Certainly if (0.7) holds, thenWhen A is semisimple there are three equivalent definitions of the socle: the sum of the minimal left ideals, the sum of the minimal right ideals, and the linear subspace generated by the "rank one" elements. These in turn essentially reduce

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On regularities and Fredholm theory

Cited by 12 publications

References 15 publications

Fredholm theory in ordered Banach algebras

Fredholm theory in ordered Banach algebras

r-FREDHOLM THEORY IN BANACH ALGEBRAS

Adjugates in Banach algebras

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