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In this paper, for a bounded linear operator defined on a complex Banach space of infinite dimension, we consider the set of isolated points in its approximate point spectrum, which are eigenvalues of finite multiplicity; this set can be equal to the spectrum of the operator but without its upper semi-Fredholm spectrum, and this relation or equality defines in the literature a new spectral property called the property (ωπ) and is a generalization of the classical a-Weyl theorem. We establish some characterizations and consequences about the property (ωπ), some with topological aspects. Furthermore, we study this property through the Riesz functional calculus. Part of the spectral structure of a linear operator verifying property (ωπ) is described, obtaining some associated properties. [ABSTRACT FROM AUTHOR]

In this paper, for a bounded linear operator defined on a complex Banach space of infinite dimension, we consider the set of isolated points in its approximate point spectrum, which are eigenvalues of finite multiplicity; this set can be equal to the spectrum of the operator but without its upper semi-Fredholm spectrum, and this relation or equality defines in the literature a new spectral property called the property $ (\omega{ \pi }) $ and is a generalization of the classical a-Weyl theorem. We establish some characterizations and consequences about the property $ (\omega{ \pi }) $, some with topological aspects. Furthermore, we study this property through the Riesz functional calculus. Part of the spectral structure of a linear operator verifying property $ (\omega{ \pi }) $ is described, obtaining some associated properties.

In this paper, we study some local spectral properties of 3 x 3-block matrix linear relations. More precisely, we give a necessary and sufficient conditions for linear relations matrix to have the single valued property in terms of the surjective spectrum and the analytic residuum of its diagonal entries. On the following, by means of it, we illustrate a new characterization of spectra and local spectra of linear relations matrix. [ABSTRACT FROM AUTHOR]

In this paper the notions of left and right generalized Drazin–Riesz invertible linear relations are introduced and studied. For these classes of linear relations we give several characterizations which covered the ones proved in Zivkovic-Zlatanovic and Cvetkovic (Linear Multilinear Algebra 65(6):1171–1193, 2017) in the particular case of bounded operators. Some characterizations are essentially expressed by means of a Kato–Riesz decomposition and an associated projection. Among other things, we show that these linear relations are completely characterized in terms of an algebraic decomposition with a semi-Browder linear relation and a bounded Riesz operator. Finally, the application of some obtained results is given. [ABSTRACT FROM AUTHOR]

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]

The classes of bounded operators on Banach spaces which satisfy a-Weyl's theorem, Weyl's theorem and the property (w) were studied by many authors. In this paper, we introduce and study the above three Weyl's type theorems in the general context of multivalued linear operators. We begin by giving a variety of characterizations of the SVEP at a point of Kato finite type linear relations. Such characterizations will be the key to investigate the classes of linear relations which satisfy a-Weyl's theorem, Weyl's theorem and the property (w). As an application of the obtained results we present an example of linear relation which obeys a-Weyl's theorem, Weyl's theorem and it also has the property (w). We also give an example of linear relation such that the property (w), a-Weyl's theorem and Weyl's theorem fail for it. [ABSTRACT FROM AUTHOR]

We introduce the class of (M, k)-quasi-*-paranormal operators on a Hilbert space H. This class extends the classes of *-paranormal and k -quasi-*-paranormal operators. An operator T on a complex Hilbert space is called (M, k) -quasi-*-paranormal if there exists M > 0 such that ... for all x ∈ H. In the present article, we give operator matrix representation of a (M, k) -quasi-*-paranormal operator. The compactness, the invariant subspace, and some topological properties of this class of operators are studied. Several properties of this class of operators are also presented. [ABSTRACT FROM AUTHOR]

The issue of the single-valued extension property (abbreviated SVEP) for linear operators introduced by Aiena (Fredholm and local spectral theory, with applications to multipliers, Kluwer Academic Publishers, Dordrecht, 2004) and (Fredholm and local spectral theory II. With application to Weyl-type theorems, Springer, Cham, 2018), motivates several authors to develope this notion for a block linear operators matrices. In this paper, enlightened by the study of Ammar et al. (Mediterr J Math 18(2):1–27, 2021), we investigate a few properties of the local spectra of a 2 × 2 block matrix of linear relations. Besides, by the new sets originating from the SVEP, we give the necessary and sufficient conditions to characterize the spectra and the essential spectra. [ABSTRACT FROM AUTHOR]

Let be an infinite dimensional complex Hilbert space and be the algebra of all bounded linear operators on. For , we say T satisfies property (ω) if σa(T) \ σea(T) = π00(T), where π00(T) = {λ ∈ isoσ(T) : 0 < n(T − λI) < ∞}. In this paper, we research on the property (ω) for functions of operators by using the new spectrum σvea(T) which is a variant of the Weyl essential approximate point spectrum. At the same time, the stability of SVEP as well as the relationships between the two parts is also given. [ABSTRACT FROM AUTHOR]

A Banach space X is called Hopfian, if any bounded linear operator surjective is bijective. The existence of the Banach Hopfians spaces in infinite dimension was established by Gowers and Maury in 1993. In this note we obtain some characterizations of Banach spaces Hopfians by properties of the algebra of bounded linear operators B(X). [ABSTRACT FROM AUTHOR]

Let T be a linear bounded operator on a complex Banach space X. In this paper, we introduce a local version of the Ritt resolvent condition [LR] for Banach space operators T. We start by showing that this concept is weaker than the classical Ritt condition [R]. We prove that, for operators with single-valued extension property (SVEP), estimate [LR] extends, with a larger constant, to some sector Kδ. Moreover, by extending some Ritt's theorems to the local case for operators with the SVEP, several characterizations of the local sublinear decay of Tn -Tn+1 have been established. [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]

We introduce the class of (M; k)-quasi-parahyponormal operators on a separable Hilbert space as an extension of the classes of parahyponormal and k-quasiparahyponormal operators given in [13]. The matrix representation of an (M; k)-quasiparahyponormal operator, the finite ascent, the SVEP and other several properties of such class of operators are also presented. [ABSTRACT FROM AUTHOR]

Let 𝒯 = (Tt)t≥0 be a C0-semigroup on a separable infinite dimensional Banach space X, with generator A. In this paper, we study the relationship between the single valued extension property of generator A, and the M-hypercyclicity of the C0-semigroup. Specifically, we prove that if A does not have the single valued extension property at λ ∈ iℝ, then there exists a closed subspace M of X, such that the C0-semigroup 𝒯 is M-hypercyclic. As a corollary, we get certain conditions of the generator A, for the C0-semigroup to be M-hypercyclic.

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]

In this article, a-Browder's classical theorem is considered through the property (B v) , and we show that if two operators are norm equivalent, then property (B v) holds for one if and only if it holds for the other. The necessary conditions for the difference of the spectrum and essential approximate point spectrum of the tensor product of two operators coincide with the product of differences between the spectrum and the essential approximate point spectrum of its two factors are investigated. We also discuss the necessary conditions for the tensor product of two operators to verify the property (B v) and simultaneously give the equivalence between their various spectra with their Browder spectrum, likewise with the Drazin spectrum. [ABSTRACT FROM AUTHOR]

In this article, for a bounded linear operator defined on a complex infinite-dimensional Banach space, through classical methods of local spectral theory, we study an important variant of a-Browder's theorem, the variant known as the property (az). Among other new results, are obtained some characterization with topological terms showing that a set of operators that verify this property (az) is closed. Some existing results are generalized, and from a new perspective, the stability of this property (az) is studied under the classic perturbations that commute with the operator. [ABSTRACT FROM AUTHOR]

Let (T (t)) t ≥ 0 be a strongly continuous semigroup of operators on a Banach space X and A be its generator. We show, in this paper, the following results: σ ∗ (T (t)) \ { 0 } = exp (t σ ∗ (A)) , for t > 0 , where σ ∗ runs over a family of some usual spectra. [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 article, a-Browder’s classical theorem is considered through the property (Bv), and we show that if two operators are norm equivalent, then property (Bv) holds for one if and only if it holds for the other. The necessary conditions for the difference of the spectrum and essential approximate point spectrum of the tensor product of two operators coincide with the product of differences between the spectrum and the essential approximate point spectrum of its two factors are investigated. We also discuss the necessary conditions for the tensor product of two operators to verify the property (Bv) and simultaneously give the equivalence between their various spectra with their Browder spectrum, likewise with the Drazin spectrum.

Previous studies have demonstrated that orientation-specific deprivation in early life can lead to neural deficits of spatial vision in certain space, and can even result in meridional amblyopia (MA). Individuals with astigmatism are the optimal and natural models for exploring this asymmetric development of spatial vision in the human visual system. This study aims to assess the contrast sensitivity function (CSF) and EEG signals along two principal meridians in participants with regular astigmatism when being optimal optical corrected. Twelve participants with astigmatism (AST group, 20 eyes) and thirteen participants with (MA group, 19 eyes) were recruited in the current study. CSFs and spatial sweep visual evoked potentials (sVEP) were measured with vertical and horizontal sinewave gratings along two principal meridians monocularly. Area under log CSF (AULCSF), spatial frequency threshold corresponding to 80% contrast gratings (SF threshold at 80% ctr), and CSF acuity were calculated from CSF test. In addition, sVEP amplitudes and thresholds were calculated with the recursive least square method. Participants with astigmatism exhibited marked vertical-horizontal resolution disparities even after they were corrected with optimal optical corrections. CSF tests showed that AULCSF along weak meridian (measured with horizontal gratings) was lower than that along strong meridian (measured with vertical gratings) in both groups. Significant meridional disparity of CSF acuity was also found in both groups. In addition, the MA group showed larger meridional disparity compared to the AST group. Spatial sVEP thresholds also supported the existence of marked meridional disparity. Our results suggest that meridian-specific partial deprivation in early life might lead to monocularly asymmetric development of spatial vision in the human visual system. In terms of application, we tested the feasibility and reliability of adopting psychophysical and EEG scalings to investigate the asymmetric development of spatial vision related to astigmatism. These paradigms are potentially applicable to reduce and even eliminate the meridional disparity in the primary visual cortex by adopting perceptual learning or other vision-related interventions.

Previous studies have demonstrated that orientation-specific deprivation in early life can lead to neural deficits of spatial vision in certain space, and can even result in meridional amblyopia (MA). Individuals with astigmatism are the optimal and natural models for exploring this asymmetric development of spatial vision in the human visual system. This study aims to assess the contrast sensitivity function (CSF) and EEG signals along two principal meridians in participants with regular astigmatism when being optimal optical corrected. Twelve participants with astigmatism (AST group, 20 eyes) and thirteen participants with (MA group, 19 eyes) were recruited in the current study. CSFs and spatial sweep visual evoked potentials (sVEP) were measured with vertical and horizontal sinewave gratings along two principal meridians monocularly. Area under log CSF (AULCSF), spatial frequency threshold corresponding to 80% contrast gratings (SF threshold at 80% ctr), and CSF acuity were calculated from CSF test. In addition, sVEP amplitudes and thresholds were calculated with the recursive least square method. Participants with astigmatism exhibited marked vertical-horizontal resolution disparities even after they were corrected with optimal optical corrections. CSF tests showed that AULCSF along weak meridian (measured with horizontal gratings) was lower than that along strong meridian (measured with vertical gratings) in both groups. Significant meridional disparity of CSF acuity was also found in both groups. In addition, the MA group showed larger meridional disparity compared to the AST group. Spatial sVEP thresholds also supported the existence of marked meridional disparity. Our results suggest that meridian-specific partial deprivation in early life might lead to monocularly asymmetric development of spatial vision in the human visual system. In terms of application, we tested the feasibility and reliability of adopting psychophysical and EEG scalings to investigate the asymmetric development of spatial vision related to astigmatism. These paradigms are potentially applicable to reduce and even eliminate the meridional disparity in the primary visual cortex by adopting perceptual learning or other vision-related interventions. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of Weyl's theorem and the stability of SVEP for 3 × 3 upper triangular operator matrices, respectively. Also, we characterize the equivalent conditions when the stabilities of both Weyl's theorem and SVEP for 3 × 3 upper triangular operator matrices hold at the same time. [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]

In this paper, we introduce a polynomially normal operator on a complex Hilbert space, extending the notation of n-normal and normal operators. Several basic properties of polynomially normal operator are firstly presented. We show some spectral properties of polynomially normal operators under new assumption in literature. Precisely, we prove that σ (T) = σ a (T) , ker (T - z) ⊥ ker (T - w) if z and w are distinct eigen-values of T and others results. Thus, we generalize some results for n-normal and normal operators. [ABSTRACT FROM AUTHOR]

In this paper we explore some characteristics of the quasi-Fredholm resolvent set ρ qf (T) of an operator T defined on an infinite dimensional Banach space X. Moreover, in the case of Hilbert space H, we study the stability of the SVEP and describe the operators for which the SVEP is preserved under compact perturbations using quasi-Fredholm spectrum and ρ qf (T) . [ABSTRACT FROM AUTHOR]

Property (UWΠ), introduced in Berkani and Kachad (BullKorean Math Soc 49:1027-1040, 2015) and studied more recently in Aiena and Kachad (Acta Sci Math (Szeged) 84:555-571, 2018) may be thought as a variant of Browder's theorem, or Weyl's theorem, for bounded linear operators acting on Banach spaces. In this article we study the stability of this property under some commuting perturbations, as quasinilpotent perturbation and, more in general, under Riesz commuting perturbations. We also study the transmission of property (UWΠ) from T to f (T), where f is an analytic function defined on a neighborhood of the spectrum of T . Furthermore, it is shown that this property is transferred from a Drazin invertible operator T to its Drazin inverse S. [ABSTRACT FROM AUTHOR]

In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the. [ABSTRACT FROM AUTHOR]

Schmoeger has shown that if Weyl’s theorem holds for an isoloid Banach space operator T ∈ B(X) with stable index, then it holds for f (T ) whenever f ∈ Holo σ(T ) is a function holomorphic on some neighbourhood of the spectrum of T . In this note we establish a converse.

Given Hilbert space operators such that ( the closure of the numerical range of S), the similarities for invertible A and have been considered by a number of authors over past few decades. A classical result of C. R. De Prima (resp., I. H. Sheth) says that if A and are normaloid or convexoid (resp., A is hyponormal), then implies A is unitary (resp., implies A is self-adjoint). This paper uses (Putnam-Fuglede theorem type) commutativity results to obtain generalizations of extant results on similarities of the above type. Amongst other results, it is proved that if with A invertible and , then: (i) A normaloid implies either A is unitary or ; (ii) operators A satisfying the positivity condition are unitary. If the operator A in (resp., ) is w-hyponormal or class with , then a sufficient condition for A to be unitary (resp., A to be self-adjoint) is that ; furthermore, one may drop the hypothesis in the case in which . [ABSTRACT FROM AUTHOR]

A bounded operator T in a Banach space X is said to satisfy the descent spectrum equality, if the descent spectrum of T as an operator on X coincides with the descent spectrum of T as an element of the algebra B (X) of all bounded linear operators on X. In this paper, we give some conditions under which the equality σ desc (T) = σ desc (T , B (X)) holds for a single operator T. [ABSTRACT FROM AUTHOR]

Property (UWII) for a bounded linear operator T ϵ L(X) on a Banach space X is a variant of Browder's theorem, and means that the points λ of the approximate point spectrum for which λI - T is upper semi-Weyl are exactly the spectral points λ such that λI - T is Drazin invertible. In this paper we investigate this property, and we give several characterizations of it by using typical tools from local spectral theory. We also relate this property with some other variants of Browder's theorem (or Weyl's theorem). [ABSTRACT FROM AUTHOR]

A Banach space operator S satisfies property (aw) if σ(S) \ σw(S) = E0 a (S), where E0 a(S) is the set of all isolated point in the approximate point spectrum which are eigenvalues of finite multiplicity. Property (aw) does not transfer from operators A and B to their tensor product A ⊗ B, so we give necessary and/or sufficient conditions ensuring the passage of property (aw) from A and B to A ⊗ B. Perturbations by Riesz operators are considered. [ABSTRACT FROM AUTHOR]

Let Mn(C) denote the algebra of all n × n complex matrices, and fix a nonzero vector x0 in Cn. For any matrix T ∈Mn(C), let σT (x0) denote the local spectrum of T at x0. Given three scalars μ, ν and ξ simultaneously nonzero, we study maps ϕ on Mn(C) satisfying σμSTS+νTS+ξST (x0) = σμϕ(S)ϕ(T)ϕ(S)+νϕ(T)ϕ(S)+ξϕ(S)ϕ(T)(x0) for all S, T ∈Mn(C). Our main result extends and unifies the main results of several papers on maps on Mn(C) preserving the local spectrum of different products. [ABSTRACT FROM AUTHOR]

Let Mn(ℂ) denote the algebra of all n ⨰ n complex matrices, and fix a nonzero vector x0 in ℂn. For any matrix T ∊Mn(ℂ), let σT (x0) denote the local spectrum of T at x0. Given three scalars α, V and ξ simultaneously nonzero, we study maps φ on Mn(ℂ) satisfying σαSTS+VTS+ξST (x0)= σαφ(S)φ(T)φ(S)+Vφ(T)φ(S)+ξφ(S)φ(T)(x0) for all S, T ∊Mn(ℂ). Our main result extends and unifies the main results of several papers on maps on Mn(ℂ) preserving the local spectrum of different products. [ABSTRACT FROM AUTHOR]

A Banach space operator A∈B(X) is polaroid, A∈(P), if the isolated points of the spectrum σ(A) are poles of the operator; A is hereditarily polaroid, A∈(HP), if every restriction of A to a closed invariant subspace is polaroid. It is seen that operators A∈(HP) have SVEP - the single-valued extension property - on Φsf(A)={λ:A-λ is semi Fredholm }. Hence Φ+sf(A)={λ∈Φsf(A),\ind(A-λ)>0}=⊘ for operators A∈(HP), and a necessary and sufficient condition for the perturbation A+K of an operator A∈B(X) by a compact operator K\inB(X) to be hereditarily polaroid is that Φ+sf(A)=⊘. A sufficient condition for A∈B(X) to have SVEP on Φsf(A) is that its component Ωa(A)={λ∈Φsf(A):\ind(A-λ)≤0} is connected. We prove: If A∈B(H) is a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator K∈B(H) such that A+K∈(HP) is that Ωa(A) is connected. [ABSTRACT FROM AUTHOR]

For nonnegative integers $$n$$ and $$k$$ , we introduce in this paper a new class of $$(n,k)$$ -quasi- $$*$$ -paranormal operators satisfying This class includes the class of $$n$$ - $$*$$ -paranormal operators and the class of $$(1,k)$$ -quasi- $$*$$ -paranormal operators contains the class of $$k$$ -quasi- $$*$$ -class $$A$$ operators. We study the basic properties of $$(n,k)$$ -quasi- $$*$$ -paranormal operators, like relations of this new class of operators with other classes known in the literature, their matrix representation, and properties of their spectra. [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]

We introduce the class of -quasi--isometric operators on Hilbert space. This generalizes the class of -isometric operators on Hilbert space introduced by Agler and Stankus. An operator is said to be -quasi--isometric if In this paper matrix representation of a -quasi--isometric operator is given. Using this representation we establish some basic properties of this class of 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]