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For a bounded linear operator, by local spectral theory methods, we study the property (b z) , which means that the difference of the approximate point spectrum with the upper semi-Fredholm spectrum coincides with the set of all finite-range left poles. We will investigate this property under closed proper subspaces of X , also under the tensor product. In addition, the relationships of this property with other spectral properties are studied. Among others, we will obtain several characterizations for the operators that verify the property (bz) and show that the set of these operators is closed. [ABSTRACT FROM AUTHOR]

In this paper, we give necessary and sufficient conditions for a bounded linear operator T to be generalized Drazin-Riesz invertible or generalized Drazin-meromorphic invertible. Also, we study Browder type theorems by means of the set of interior points of various parts of spectrum of T. [ABSTRACT FROM AUTHOR]

Property (U W Π) for a bounded linear operator T ∈ L (X) on a Banach space X means that the points λ of the approximate point spectrum for which λ I - T is upper semi-Weyl are exactly the poles of the resolvent of T. In this paper we give some new results on this property and relate it with some other variants of Browder's theorem, as property (gaz) and property (Z Π a) . In the last part the theory is applied to Toeplitz operators T ϕ with symbol ϕ on Hardy spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the properties (b z) and (w π 00 a ) , which we had introduced in [Ben Ouidren K, Zariouh H. New approach to a-Weyl's theorem and some preservation results. Rend Circ Mat Palermo. doi: ], for an operator having the SVEP on the complementary of distinguished parts of its spectrum. We prove in particular, that a bounded linear operator T acting on a Banach space has the SVEP on the complementary of σ u f (T) if and only if T possesses property (b z). We also study the stability of these properties under several commuting perturbations, and we prove that if T possesses property (b z) then f (T) + R possesses property (b z) for every Riesz operator R commuting with T and f ∈ Hol (σ (T)). We also give an example which shows that the property (w π 00 a ) is generally unstable under this type of perturbations, and we prove that if T possesses property (w π 00 a ) then T + R possesses property (w π 00 a ) ⇔ π 00 a (T + R) ∩ σ a (T) ⊂ π 00 a (T). Analogous results are proved for the property (g w π 00 a ) and some applications to the class of a-isoloid-type operators are given. [ABSTRACT FROM AUTHOR]

Let { T (t) } t ≥ 0 be a C 0 -semigroup of bounded linear operators on the Banach space X into itself and let A be their infinitesimal generator. In this paper, we show that if T(t) is uniformly ergodic, then A does not have the single valued extension property, which implies that A must have a non-empty interior of the point spectrum. Furthermore, we introduce the local mean ergodic for C 0 -semigroup T(t) at a vector x ∈ X and we establish some conditions implying that T(t) is a local mean ergodic at x. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of property (BR) and property (BgR) for bounded linear operators defined on an infinite-dimensional Banach space. These properties in connection with Weyl type theorems and in the frame of polaroid operators are investigated. Moreover, we study the stability of these properties under perturbations by commuting finite-dimensional, quasi-nilpotent, Riesz and algebraic operators. [ABSTRACT FROM AUTHOR]

We discuss some relations between the local existence of analytic selections of eigenvectors (LSP $$=$$ 'NON-SVEP') for an operator in Banach space and some chaoticity properties of linear dynamical system (with discrete or continuous time) generated by this operator. Our main goal is to prove the existence of a strong connection of some results known for many years in the local spectral theory to some important problems (which seem to be not solved so far) in the linear chaos theory. We also find a simple particular solution of the problem formulated in 'Eigenvectors Selection Conjecture' (Banasiak and Moszyński in Discrete Contin Dyn Syst A 20(3):577-587, 2008, Conjecture 4.3, p. 585) and we formulate a new convenient spectral criterion for linear chaos. To make the assumptions more clear we introduce some special parts of the point spectrum of a closed operator, including the right-inversion point spectrum. Using this new criterion we prove chaoticity of a large class of super-upper-triangular operators in $$l^p$$ and $$c_0$$ spaces and also of some strongly continuous semi-groups generated by such operators. [ABSTRACT FROM AUTHOR]

The paper introduces the notion of properties $$(Z_{\Pi _{a}})$$ and $$(Z_{E_{a}})$$ as variants of Weyl's theorem and Browder's theorem for bounded linear operators acting on infinite dimensional Banach spaces. A characterization of these properties in terms of localized single valued extension property is given, and the perturbation by commuting Riesz operators is also studied. Classes of operators are considered as illustrating examples. [ABSTRACT FROM AUTHOR]