Ruston, Riesz and perturbation classes

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].…”

Polynomially Riesz perturbations II

“…An operator A ∈ B(X) is called polynomially Riesz if A ∈ Poly −1 (R(X)). We recall the following result [20,Theorem 11.1], [22,Theorem 2.3].…”

“…Our discussion about perturbation by polynomially Riesz elements of a Banach algebra concerning the set of (left, right) Fredholm elements was started in [20] (Theorems 11.2 and 12.3). This discussion is continued in [21] where we focused on the Banach algebra B(X) and investigated perturbations of left(right) Fredholm, Weyl and Browder operators by polynomially Riesz operators using the concept of communicating operators, and, specially, perturbations of some shifts were considered.…”

On polynomially Riesz operators

“…These operators have been recently discussed in [20]. Some parts of the following theorem are contained in Theorem 11.1 in [19]. For the sake of completeness we give the whole proof and start with the following simple lemma.…”