性质(gz)与广义Weyl型定理
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摘要
称有界线性算子T∈L(X)满足性质(gz),如果T的上半B-Weyl谱在T的谱集中的补集恰好为T的逼近点谱中孤立的特征值全体.本文首先讨论了性质(gz)与其它广义Weyl型定理之间的关系;然后利用新定义的谱集σ_2(T)与Drazin谱之间的关系,给出了Banach空间中有界线性算子T及其函数演算满足性质(gz)的等价刻画;最后利用所得结论讨论了弱-H(P)类算子的性质(gz).
A bounded linear operator T ∈ L(X) defined on a Banach space X satisfies property(gz), if the complement in the spectrum σ(T) of the upper semi-B-Weyl spectrumσ_(SBF)~-_+(T) is the set of all isolated points of the approximate point spectrum σ_a(T) which are eigenvalues. In this note, we first study the conditions between property(gz) and other generalized Weyl-type theorem, then establish for a bounded linear operator and the calculus defined on a Banach space the sufficient and necessary conditions for which property(gz)holds by means of the condition between the new spectrum σ_2(T) and the Drazin spectrum.In addition, using the main result, property(gz) of the class of weak-H(P) is considered.
A bounded linear operator T ∈ L(X) defined on a Banach space X satisfies property(gz), if the complement in the spectrum σ(T) of the upper semi-B-Weyl spectrumσ_(SBF)~-_+(T) is the set of all isolated points of the approximate point spectrum σ_a(T) which are eigenvalues. In this note, we first study the conditions between property(gz) and other generalized Weyl-type theorem, then establish for a bounded linear operator and the calculus defined on a Banach space the sufficient and necessary conditions for which property(gz)holds by means of the condition between the new spectrum σ_2(T) and the Drazin spectrum.In addition, using the main result, property(gz) of the class of weak-H(P) is considered.
引文
[1]Weyl H V.Uber beschrankte quadratische formen,deren differenz vollstetig ist.Rendiconti Del Circolo Matematico Di Palermo,1909,27(1):373-392
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[3]Rakoc evicV.Operators obeying a-Weyl's theorem.Romanian Journal of Pure and Applied Mathematics,1989,34(10):915-919
[4]Berkani M,Koliha J J.Weyl type theorems for bounded linear operators.Acta Scientiarum Mathematicarum,2003,69(1):359-376
[5]Amouch M,Berkani M.On the Property(gω).Mediterranean Journal of Mathematics,2008,5(3):371-378
[6]Zariouh H.Property(gz)for bounded linear operators.Matematicki Vesnik,2013,65(1):94-103
[7]Aiena P.Fredholm and Local Spectral Theory,with Applications to Multipliers.Springer Netherlands,2004
[8]Grabiner S.Uniform ascent and descent of bounded operators.Journal of the Mathematical Society of Japan,1982,34(2):172-175
[9]Taylor A E.Theorems on ascent,descent,nullity and defect of linear operators.Mathematische Annalen,1966,163(1):18-49
[10]Oberai K K.On the Weyl spectrum,II.Illinois Journal of Mathematics,1977,21(1):84-90
[11]Berkani M,Zariouh H.Generalized a-weyl's theorem and perturbations.Functional Analysis,Approximation and Computation,2010,2(1):7-18
[12]戴磊,曹小红,孙晨辉.广义(ω)性质的一个注记.数学学报,2010,53(2):219-226(Dai L,Cao X H,Sun C H.A Note on Generalized Property(ω).Acta Mathematica Sinica,2010,53(2):219-226)
[13]Berkani M.B-Weyl spectrum and poles of the resolvent.Journal of Mathematical Analysis and Applications,2002,272(2):596-603
[14]Cao X H,Liu A F.Generalized Kato type operators and property(ω)under perturbations.Linear Algebra and Its Applications,2012,436(7):2231-2239
[15]Merkhta M.Sur la Théorie Spectrale Locale et Limite des Nilpotents.Proceedings of the American Mathematical Society,1990,110(3):621-631
[16]孙晨辉,曹小红,戴磊.Weyl型定理及其摄动.数学学报,2009,52(1):73-80(Sun C H,Cao X H,Dai L.Weyl Type Theorem and Perturbations.Acta Mathematica Sinica,2009,52(1):73-80)
[17]Coburn L A.Weyl's Theorem for Nonnormal Operators.Michigan Mathematical Journal,1966,13(3):285-288
[18]Duggal B P,Djordjevic S V.Dunford's property(C)and Weyl's theorems.Integral Equations and Operator Theory,2002,43(3):290-297
[19]Chen L,Ran Y B,Zhang Y.p-hyponormal operators are subscalar.Proceedings of the American Mathematical Society,2003,131(9):2753-2759
[20]Oudghiri M.Weyl's and Browder's theorems satisfying the SVEP.Studia Mathematica,2004,163(1):85-101
[2]Rakocevic V.On a class of operators.Matematicki Vesnik,1985,37(4):423-426
[3]Rakoc evicV.Operators obeying a-Weyl's theorem.Romanian Journal of Pure and Applied Mathematics,1989,34(10):915-919
[4]Berkani M,Koliha J J.Weyl type theorems for bounded linear operators.Acta Scientiarum Mathematicarum,2003,69(1):359-376
[5]Amouch M,Berkani M.On the Property(gω).Mediterranean Journal of Mathematics,2008,5(3):371-378
[6]Zariouh H.Property(gz)for bounded linear operators.Matematicki Vesnik,2013,65(1):94-103
[7]Aiena P.Fredholm and Local Spectral Theory,with Applications to Multipliers.Springer Netherlands,2004
[8]Grabiner S.Uniform ascent and descent of bounded operators.Journal of the Mathematical Society of Japan,1982,34(2):172-175
[9]Taylor A E.Theorems on ascent,descent,nullity and defect of linear operators.Mathematische Annalen,1966,163(1):18-49
[10]Oberai K K.On the Weyl spectrum,II.Illinois Journal of Mathematics,1977,21(1):84-90
[11]Berkani M,Zariouh H.Generalized a-weyl's theorem and perturbations.Functional Analysis,Approximation and Computation,2010,2(1):7-18
[12]戴磊,曹小红,孙晨辉.广义(ω)性质的一个注记.数学学报,2010,53(2):219-226(Dai L,Cao X H,Sun C H.A Note on Generalized Property(ω).Acta Mathematica Sinica,2010,53(2):219-226)
[13]Berkani M.B-Weyl spectrum and poles of the resolvent.Journal of Mathematical Analysis and Applications,2002,272(2):596-603
[14]Cao X H,Liu A F.Generalized Kato type operators and property(ω)under perturbations.Linear Algebra and Its Applications,2012,436(7):2231-2239
[15]Merkhta M.Sur la Théorie Spectrale Locale et Limite des Nilpotents.Proceedings of the American Mathematical Society,1990,110(3):621-631
[16]孙晨辉,曹小红,戴磊.Weyl型定理及其摄动.数学学报,2009,52(1):73-80(Sun C H,Cao X H,Dai L.Weyl Type Theorem and Perturbations.Acta Mathematica Sinica,2009,52(1):73-80)
[17]Coburn L A.Weyl's Theorem for Nonnormal Operators.Michigan Mathematical Journal,1966,13(3):285-288
[18]Duggal B P,Djordjevic S V.Dunford's property(C)and Weyl's theorems.Integral Equations and Operator Theory,2002,43(3):290-297
[19]Chen L,Ran Y B,Zhang Y.p-hyponormal operators are subscalar.Proceedings of the American Mathematical Society,2003,131(9):2753-2759
[20]Oudghiri M.Weyl's and Browder's theorems satisfying the SVEP.Studia Mathematica,2004,163(1):85-101