Weyl Type Theorems And Related Topics

In1909, Weyl studied the spectrum of all compact perturbations of a self adjointoperator and showed thatfor each self adjoint operator T, where σ_w（T） denotes the Weyl spectrum of T andπ₀0（T） denotes the set of all isolated eigenvalues of T with finite multiplicity. Given T∈B（H）, following Coburn, we say that Weyl’s theorem holds for T if T satisfies the aboveequation. Now Weyl’s theorem has been extended to other classes of operators, suchas hyponormal operators, p-hyponormal operators, algebraic hyponormal operatorsand so on. People also study Weyl’s theorem under some perturbations and analyticfunctional calculus. People also study some generalizations of Weyl’s theorem. Tointroduce Weyl type theorems, we give some necessary definitions.In this paper,C and N denote the set of complex numbers and the set of positiveintegers, respectively. We let H denote a complex separable infinite dimensional Hilbertspace. In addition, we let B（H） denote the set of all bounded linear operators on H,and let K（H） denote the ideal of all compact operators in B（H）.Let T∈B（H）. We denote by σ_a（T） and σs（T） the approximate points spectrumof T and the surjective spectrum of T respectively. We denote by ker（T） and R（T） thekernel of T and the range of T respectively. We say T is a semi-Fredholm operator, ifR（T） is closed and either dim ker（T） or dim ker（T） is finite. If T is a semi-Fredholmoperator, we call ind （T）=dim ker（T） dim ker（T） the index of T. In addition, if∞<ind （T）<+∞, T is called a Fredholm operator. In particular, if T is aFredholm operator of index0, we call T is a Weyl operator.For T∈B（H） and n≥0, we define T[n]as the restriction of T to R（Tn） viewedas a map from R（Tn） into R（Tn）. If for some n the space R（Tn） is closed and T[n]is aFredholm operator, we call T is a B-Fredholm operator. In this case, it is well knownthat T[m]is Fredhom operator and ind （T[m]）=ind （T[n]） for all m≥n. This meansone can define the index of a B-Fredholm operator T as the index of T[n], where n is anynonnegative number such that R（Tn） is closed and T[n]is a Fredholm operator. Wesay T is a B-Weyl operator if T is a B-Fredholm of index zero. The B-Weyl spectrumσBW（T） of T is defined asσBW（T）:={λ∈C: T λ is not B-Weyl}.The Wolf spectrum σlre（T）, the essential spectrum σe（T） and the Weyl spectrumσw（T） are defined as:σlre（T）:={λ∈C: T λ is not semi-Fredholm},σe（T）:={λ∈C: T λ is not Fredholm},andσw（T）:={λ∈C: T λ is not Weyl},respectively. It is easy to see that σlre（T） σe（T） σw（T）. ρs F（T）:=C\σlre（T） iscalled the semi-Fredholm domain of T. We denoteρ0s F（T）:={λ∈ρs F（T）: T λ is Weyl},ρ+s F（T）:={λ∈ρs F（T）: ind （T λ）>0},andρ s F（T）:={λ∈ρs F（T）: ind （T λ）<0},respectively. The essential approximate points of T is defined asσea（T）:=σa（T+K）.K∈K（H） It is easy to check that σea（T）=σlre（T）∪ρ+s F（T）. Given σ C, we denote by iso σall isolated points of σ.In1984, Rakoˇcevi′c generalized the notion of Weyl’s theorem and defined a-Weyl’stheorem. We say that T∈B（H） satisfies a-Weyl’s thereom, denoted by T∈（aW）, ifσa（T）\σea（T）=πa00（T）, where πa00（T）:={λ∈iso σa（T）:0<dim ker（T λ）<∞}.In1985, Rakoˇcevi′c gave another generalization of Weyl’s theorem, which namedproperty （ω）. We say T∈B（H） has property （ω）, denoted by T∈（ω）, if σa（T）\σea（T）=π00（T）.In2003, Berkani and Koliha generalized the notion of Weyl’s theorem, we say thatgeneralized Weyl’s theorem holds for T∈B（H）, denoted by T∈（gW）, if there is theequality σ（T）\σBW（T）=E（T）, where E（T）=σp（T）∩iso σ（T）. This is anothergeneralization of the classical Weyl’s theorem.Now there is a lot work on Weyl type theorems. Weyl type theorems are extendedto many important classes of operators. People also study the stability of Weyl typetheorems under commuting finite rank perturbations, nilpotent perturbations and otherperturbations. In addition, there is a lot work on the stability of Weyl type theoremsunder analytic functional calculus. We shall study Weyl type theorems under smallcompact perturbations and analytic functional calculus.Polaroid property is closed related to Weyl type theorems, which is an importantcondition to study Weyl type theorems. Now we are going to introduce polaroid typeproperties. The ascent of T is defined as the smallest non-negative integer p:=p（T）such that ker（Tp）=ker（Tp+1）. If such number does not exist, we define p（T）=∞.On the other hand, the descent of T is defined as the smallest non-negative integerq:=q（T） such that R（Tq）=R（Tq+1）. If such number does not exist, we defineq（T）=∞. T is called Drazin invertible, if p（T） and q（T） are all finite. T is called leftDrazin invertible if p（T）<∞and R（Tp（T）+1） is closed. Analogously, T is called rightDrazin invertible, if q（T）<∞and R（Tq（T）） is closed. We say λ∈σa（T） is a left poleof T if T λ is left Drazin invertible. We say λ∈σs（T） is a right pole of T if T λ is right Drazin invertible.Following Duggal, Harte and Jeon, we say T is polaroid, denoted by T∈（P）, ifeach λ∈iso σ（T） is a pole of T.Aiena, Choˉand Gonza′lez generalized this notion and defined polaroid type proper-ties. We say T is a-polaroid, if each λ∈iso σa（T） is a pole of T, denoted by T∈（AP）.T∈B（H） is said to be left polaroid, if each λ∈iso σa（T） is a left pole of T, denotedby T∈（LP）; T∈B（H） is said to be right polaroid, if each λ∈iso σs（T） is a rightpole of T, denoted by T∈（RP）.It is well known thatT∈（AP） T∈（LP） T∈（P）andT∈（LP） T∈（RP）.Aiena and other experts studied the stability of polaroid type properties undersome commuting perturbations. We are going to study polaroid type properties andapproximation.To introduce our main results, we give some other definitions.Let T∈B（H）. If σ is a clopen subset of σ（T）, there exists an analytic Cauchydomain such that σ and [σ（T）\σ]∩=. We denote by E（σ; T） the Rieszidempotent of T respect to σ, that is,where Γ:=is positively oriented with respect to in the complex analysis sense.In this case, we denote H（σ; T）=R（E（σ; T））. In particular, if λ is an isolated pointin σ（T）, then {λ} is a clopen subset of σ（T）. We write H（λ; T） instead of H（{λ}; T）.If, in addition, dim H（λ; T）<∞, we say λ is a normal eigenvalue of T. We denoteby σ0（T） all normal eigenvalues of T. Let Hol（σ（T）） denote the set of all functions fwhich are analytic on some neighbourhood of σ（T）（the neighbourhood depends on f）. Now we are going to show our main results.Theorem1. Let T∈B（H）. Then f（T）∈（aW） for all f∈Hol（σ（T）） if and only ifthe following conditions hold.（1） T∈（aW）.（2） If ρ s F（T）=, then there exists no λ∈ρs F（T） such that0<ind （T λ）<+∞.（3） If σ0p（T）∩[ρs F（T）∪ρs F（T）]=, then iso σa（T） σp（T）.Theorem2. Let T∈B（H）. Then f（T）∈（ω） for all f∈Hol（σ（T）） if and only ifthe following conditions hold.（1） T∈（ω）.（2） If ρ s F（T）=, then σ0（T）=and there exists no λ∈ρs F（T） such that0<ind （T λ）<+∞.（3） If σ0（T）=, then iso σ（T） σp（T）.Theorem3. Given T∈B（H） and ε>0, there exists K∈K（H） with K <ε suchthat T+K∈（aW） and T∈（ω）.Theorem4. Let T∈B（H）. Then f（T）∈（gW） for all f∈Hol（σ（T）） if and only ifthe following conditions hold.（1） T∈（gW）.（2） ind （T λ）· ind （T μ）≥0, λ, μ∈/σe（T）.（3） If E（T）=, then iso σ（T） σp（T）.Theorem5. Given T∈B（H） and ε>0, there exists K∈K（H） with K <ε suchthat T+K∈（gW）.Theorem6. Given T∈B（H）. Then there exists δ>0such that T+K∈（gW） forall K∈K（H） with K <δ if and only if the following conditions hold. （1） T∈（gW）.（2）C\σw（T） consists of finite connected components.（3） iso σw（T）=.We study polaroid type properties and compact perturbations and get the follow-ing results.We denote F （T）={λ∈σlre（T）: δ>0s.t. ind （T μ）<0μ∈Bδ（λ）\{λ} and min ind （T μ）=0μ∈Bδ（λ）\[{λ}∪ρss F（T）]}.Theorem7. Given T∈B（H） and ε>0, there exists K∈K（H） with K <ε suchthat T+K∈（P）.Theorem8. Given T∈B（H）. Then the following are equivalent.（1） For ε>0, there exists K∈K（H） with K <ε such that T+K∈/（P）.（2） There exists K∈K（H） such that T+K∈/（P）.（3） iso σw（T）=.Theorem9. Given T∈B（H）. Then the following are equivalent.（1） For ε>0, there exists K∈K（H） with K <ε such that T+K∈（AP）.（2） F （T）=.Theorem10. Given T∈B（H）. Then the following are equivalent.（1） For ε>0, there exists K∈K（H） with K <ε such that T+K∈/（AP）.（2） There exists K∈K（H） such that T+K∈/（AP）.（3） iso σw（T）=or ρ s F（T）=.In addition, we study the skew symmetry of a class of operators.Some important results concerning the internal structure of complex symmetricoperators have been obtained. An efective way to investigate the structure of complex symmetric operators is to characterize which special operators are complex symmet-ric. Many classes of operators such as compact operators, weighted shifts and partialisometries are studied. However, less work has been paid to skew symmetric operators.Zagorodnyuk studied the skew symmetry of cyclic operators. Li, C. G. and Zhu, S.gave two structure theorems of skew symmetric normal operators. We characterizethe skew symmetry of a class of operators. As an application, we prove that a partialisometry T is skew symmetric if and only if the compression of T to its initial space isHamiltonian.