Polynomially Riesz perturbations

Polynomially Riesz elements

“…The following result characterizes polynomially Riesz elements and it is an improvement of Theorem 11.1 in [8]. Let us mention that, in the case when T has the strong Riesz property, the implication:…”

“…From the inclusions (12.3) in Theorem 12.1 in [8] it follows that a ∈ A is a polynomially T Riesz element if and only if there exists a function f ∈ Holo 1 (σ(a)) such that f (a) is T Riesz. Since Holo 1 (σ(a)) ⊂ Holo(σ(a)), the concept of hollomorphically Riesz element is a little more general than the concept of polynomially Riesz element.…”

“…For the equalities (2.11) and (2.12) see [8,Theorem 12.2] ((2.11) follows also from spectral mapping theorem for Fredholm spectrum and (2.12) follows also from (1.12)). 2…”

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].…”