On almost essentially Ruston elements of a Banach algebra

Abstract

We introduce the class of "almost essentially Ruston elements" with respect to a homomorphism between two Banach algebras, a class intermediate between Ruston and Fredholm elements.

Generalized B-Fredholm Banach algebra elements

“…Also it is known that the sets σ F T (a), σ W T (a) and σ B T (a) are non-empty and compact. To learn the main properties of these objects, see for example [11,12,13,14,20,10,21,8,15,1,30,22,31,32].…”

“…This well known result led to the introduction of a Fredholm theory relative to a Banach algebra homomorphism, see [11]. This theory has been developed by many authors, which have also studied other classes of objects such that Weyl, Browder and Riesz Banach algebra elements relative to a (not necessarily continuous) homomorphism, see for example [11,12,13,14,20,10,21,8,15,1,30,22,31,32].…”

Isolated spectral points and Koliha-Drazin invertible elements in quotient Banach algebras and homomorphism ranges

“…Then, Φ T (A) = {a ∈ A : T (a) ∈ B −1 } is the set of Fredholm elements relative to the homomorphism T , see [19]. Moreover, Browder, Weyl and other classes of elements relative to T have been also intoduced and studied, see for example [19,20,21,22,41,17,42,15,23,24,2,55,43,56,57]. To prove the main results of this article, it is important to recall another class of elements.…”

“…In addition, a Fredholm theory relative to Banach algebra homomorphisms was introduced in [19]. Many authors have developed this theory studying for example Fredholm, Weyl, Browder and Riesz Banach algebra element relative to homomorphisms, see for example [19,5,20,21,22,41,17,42,15,23,24,2,55,43,56,57].…”

Fredholm Theory Relative to Any Algebra Homomorphisms

“…After that, many wonderful studies for the Fredholm theory relative to homomorphisms have sprung up. Other outstanding works that you can refer are the references [13][14][15][16][17][18][19][20].…”