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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.

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]

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]

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]

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 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, 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 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]

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.

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]

In this paper we investigate the transmission of some local spectral properties from a bounded linear operator R , as SVEP, Dunford property ( C ), and property ( β ), to its Drazin inverse S , when this does exist. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of operators, called k-quasi class A n* operators, which is a superclass of hyponormal operators and a subclass of (n, k)-quasi-*-paranormal operators. We will show basic structural properties and some spectral properties of this class of operators. We show that, if T is k-quasi class An* then ... Also, we will prove Browder's theorem and a-Browders theorem for k-quasi class An* operator. [ABSTRACT FROM AUTHOR]

The paper introduces the notion of property (Bw), a version of generalized Weyl's theorem for a bounded linear operator T on an infinite dimensional Banach space X. A characterization of property (Bw) is also given. Certain conditions are explored on Hilbert space operators T and S so that T ⊕ S obeys property (Bw). [ABSTRACT FROM AUTHOR]

Abstract: The property (w) is a variant of Weyl''s theorem, for a bounded operator T acting on a Banach space. In this note we consider the preservation of property (w) under a finite rank perturbation commuting with T, whenever T is polaroid, or T has analytical core for some . The preservation of property (w) is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations. The theory is exemplified in the case of some special classes of operators. [Copyright &y& Elsevier]

Abstract: If is an upper triangular Banach space operator such that for all , then A has SVEP or satisfies (Dunford''s) condition or (Bishop''s) property or (the decomposition) property if and only if , , has the corresponding property. [Copyright &y& Elsevier]

Abstract: A Hilbert space operator T, , is totally hereditarily normaloid, , if every part and (also) every invertible part of T is normaloid. The class is large. It is proved that if a is such that the isolated eigenvalues of T are normal, then the Riesz projection P λ associated with a λ ∈iso σ(T) is self-adjoint and . [Copyright &y& Elsevier]

Abstract: Let denote the algebra of operators on a Hilbert . Let and denote the elementary operators ΔAB(X)=AXB−X and E(X)=AXB−CXD. We answer two questions posed by Turnšek [Mh. Math. 132 (2001) 349–354] to prove that: (i) if A, B are contractions, then if and only if is closed for some integer n⩾1; (ii) if A, B, C and D are normal operators such that A commutes with C and B commutes with D, then if and only if 0∈isoσ(E). [Copyright &y& Elsevier]