An operator T is said to be paranormal if ｜｜T^2 x ｜｜ 〉｜｜Tx｜｜^2 holds for every unit vector x. Several extensions of paranormal operators are considered until now, for example absolute-k-paranormal and p-paran...An operator T is said to be paranormal if ｜｜T^2 x ｜｜ 〉｜｜Tx｜｜^2 holds for every unit vector x. Several extensions of paranormal operators are considered until now, for example absolute-k-paranormal and p-paranormal introduced in [10], [14], respectively. Yamazaki and Yanagida [38] introduced the class of absolute-（p, r）-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators. An operator T ∈ B（H） is called absolute-（p, r）-paranormal operator if ｜｜｜T｜p｜T^＊ ｜^rx｜｜^r 〉 ｜｜｜T^＊｜^rx｜｜p＋r for every unit vector x ∈ H and for positive real numbers p 〉 0 and r 〉 0. The famous result of Browder, that self adjoint operators satisfy Browder＇s theorem, is extended to several classes of operators. In this paper we show that for any absolute-（p, r）- paranormal operator T, T satisfies Browder＇s theorem and a-Browder＇s theorem. It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-（p, r）-paranormal operator T, then E is self-adjoint if and only if the null space of T -μ, N（T - μ） N（T^＊ - ^μ）.展开更多
A Banach space operator satisfies generalized RakoSevi5＇s property （gw） if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the...A Banach space operator satisfies generalized RakoSevi5＇s property （gw） if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the spectrum of T. In this note, we characterize hypecyclic and supercyclic operators satisfying the property （gw）.展开更多
Two variants of the essential approximate point spectrum are discussed. We find for example that if one of them coincides with the left Drazin spectrum then the generalized a-Weyl＇s theorem holds, and conversely for ...Two variants of the essential approximate point spectrum are discussed. We find for example that if one of them coincides with the left Drazin spectrum then the generalized a-Weyl＇s theorem holds, and conversely for a-isoloid operators. We also study the generalized a-Weyl＇s theorem for Class A operators.展开更多
Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-...Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.展开更多
A∈B（H） is called Drazin invertible if A has finite ascent and descent. Let σD （A）={λ∈ C ： A -λI is not Drazin invertible } be the Drazin .spectrum. This paper shows that if Mc =（A C 0 B）is a 2 × 2 upp...A∈B（H） is called Drazin invertible if A has finite ascent and descent. Let σD （A）={λ∈ C ： A -λI is not Drazin invertible } be the Drazin .spectrum. This paper shows that if Mc =（A C 0 B）is a 2 × 2 upper triangular operator matrix acting on the Hilbert space H ＋ K, then the passage from OσD（A） U σD（B） to σD（Mc） is accomplished by removing certain open subsets of σD（A）∩σD（B） from the former, that is, there is equality σD（A）∪σD（B）=σD（MC）∪Gwhere G is the union of certain holes in σD （Me） which happen to be subsets of σD （A）∩σD （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2×2 operator matrices. By using Drazin spectrum, it also explores how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2×2 upper triangular operator matrices on the Hilbert space.展开更多
Denote a semisimple Banach algebra with an identity e by A.This paper studies the Fredholm,Weyl and Browder spectral theories in a semisimple Banach algebra,and meanwhile considers the properties of the Fredholm eleme...Denote a semisimple Banach algebra with an identity e by A.This paper studies the Fredholm,Weyl and Browder spectral theories in a semisimple Banach algebra,and meanwhile considers the properties of the Fredholm element,the Weyl element and the Browder element.Further,for a∈A,we give the Weyl’s theorem and the Browder’s theorem for a,and characterize necessary and sufficient conditions that both a and f(a)satisfy the Weyl’s theorem or the Browder’s theorem,where f is a complex-valued function analytic on a neighborhood ofσ(a).In addition,the perturbations of the Weyl’s theorem and the Browder’s theorem are investigated.展开更多
In this note we study the property （ω）, a variant of Weyl＇s theorem introduced by Rakocevic, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and suffi...In this note we study the property （ω）, a variant of Weyl＇s theorem introduced by Rakocevic, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property （ω） and approximate Weyl＇s theorem hold. As a consequence of the main result, we study the property （ω） and approximate Weyl＇s theorem for a class of operators which we call the λ-weak-H（p） operators.展开更多
基金supported by Taibah University Research Center Project(1433/803)
文摘An operator T is said to be paranormal if ｜｜T^2 x ｜｜ 〉｜｜Tx｜｜^2 holds for every unit vector x. Several extensions of paranormal operators are considered until now, for example absolute-k-paranormal and p-paranormal introduced in [10], [14], respectively. Yamazaki and Yanagida [38] introduced the class of absolute-（p, r）-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators. An operator T ∈ B（H） is called absolute-（p, r）-paranormal operator if ｜｜｜T｜p｜T^＊ ｜^rx｜｜^r 〉 ｜｜｜T^＊｜^rx｜｜p＋r for every unit vector x ∈ H and for positive real numbers p 〉 0 and r 〉 0. The famous result of Browder, that self adjoint operators satisfy Browder＇s theorem, is extended to several classes of operators. In this paper we show that for any absolute-（p, r）- paranormal operator T, T satisfies Browder＇s theorem and a-Browder＇s theorem. It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-（p, r）-paranormal operator T, then E is self-adjoint if and only if the null space of T -μ, N（T - μ） N（T^＊ - ^μ）.
文摘A Banach space operator satisfies generalized RakoSevi5＇s property （gw） if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the spectrum of T. In this note, we characterize hypecyclic and supercyclic operators satisfying the property （gw）.
文摘Two variants of the essential approximate point spectrum are discussed. We find for example that if one of them coincides with the left Drazin spectrum then the generalized a-Weyl＇s theorem holds, and conversely for a-isoloid operators. We also study the generalized a-Weyl＇s theorem for Class A operators.
文摘Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.
基金the National Natural Science Foundation of China (10571099)
文摘A∈B（H） is called Drazin invertible if A has finite ascent and descent. Let σD （A）={λ∈ C ： A -λI is not Drazin invertible } be the Drazin .spectrum. This paper shows that if Mc =（A C 0 B）is a 2 × 2 upper triangular operator matrix acting on the Hilbert space H ＋ K, then the passage from OσD（A） U σD（B） to σD（Mc） is accomplished by removing certain open subsets of σD（A）∩σD（B） from the former, that is, there is equality σD（A）∪σD（B）=σD（MC）∪Gwhere G is the union of certain holes in σD （Me） which happen to be subsets of σD （A）∩σD （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2×2 operator matrices. By using Drazin spectrum, it also explores how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2×2 upper triangular operator matrices on the Hilbert space.
文摘Denote a semisimple Banach algebra with an identity e by A.This paper studies the Fredholm,Weyl and Browder spectral theories in a semisimple Banach algebra,and meanwhile considers the properties of the Fredholm element,the Weyl element and the Browder element.Further,for a∈A,we give the Weyl’s theorem and the Browder’s theorem for a,and characterize necessary and sufficient conditions that both a and f(a)satisfy the Weyl’s theorem or the Browder’s theorem,where f is a complex-valued function analytic on a neighborhood ofσ(a).In addition,the perturbations of the Weyl’s theorem and the Browder’s theorem are investigated.
基金Supported by the Fundamental Research Funds for the Central Universities (Grant No.GK200901015)the Support Plan of the New Century Talented Person of Ministry of Education (2006)P.R. China and by Major Subject Foundation of Shanxi
文摘In this note we study the property （ω）, a variant of Weyl＇s theorem introduced by Rakocevic, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property （ω） and approximate Weyl＇s theorem hold. As a consequence of the main result, we study the property （ω） and approximate Weyl＇s theorem for a class of operators which we call the λ-weak-H（p） operators.
