Weyl's Type Theorems for Quasi-Class A Operators

A variant of Weyl theorem for a class of quasi-class A acting on an infinite complex Hilbert space were discussed. If the adjoint of T is a quasi-class A operator, then the generalized a-Weyl holds for f(T) , for every function that analytic on the spectrum of T. The generalized Weyl theorem holds for a quasi-class A was proved. Also, a characterization of the Hilbert space as a direct sum of range and kernel of a quasi-class A was given. Among other things, if the operator is a quasi-class A, then the B-Weyl spectrum satisfies the spectral theorem was characterized.


INTRODUCTION
for the Riesz points of T. Then (1.1) with the help of "Punctured neighborhoods Theorem" The ascent a: = a(T) of an operator T is the smallest non-negative integer s such that ker(T s ) = ker(T s+1 ). If such integer does not exist we put a(T) = ∞.
Analogously, the descent d = d (T) of an operator T is the smallest non-negative integer t such that ran(T t ) = ran(T t+1 ) and if such integer does not exist we put d(T) = ∞. It is well-known that if a(T) and d(T) are both finite then a(T) = d(T) [ Note that Bro(H) W(H) ⊆ , since every Fredholm operator with finite ascent and finite descent has necessary index 0, [1,9,10] . The classes of operators defined above motivate the definition of several spectra. The essential approximate point spectrum is ea a is the Browder essential approximate point spectrum. It is well-known that ea (T) { C : T I B (H)}.
Definition 2: [1] We say that a-Browder's holds for T if ea ba It is known that if T∈B(H) then a-Browder's theorem implies Browder's theorem. In [8] , the authors proved that Weyl's theorem holds for quasi-classA, in this paper, we prove that generalized Weyl's holds for quasi-class A operators.

Definition 3: An operator T∈B(H) is said to be quasiclass
The class of quasi-class A introduced and studied by Jeon and Kim [15] , for more interesting properties the reader should refer to [8,15] . Hence T 1 is one-one and onto.
Recall that an operator S∈B(H) is said to be quasiaffine transform of T (abbreviate S T ) if there is a quasiaffinity X such that XS = TX. of the spectrum. In particular, T has SVEP at every isolated point of (T) σ [16] . In [18,proposition 1.8] , Laursen proved that if T is of finite ascent, then T has SVEP.

Lemma 6: If T∈B (H) is a quasi-class A operator and S T . Then S has SVEP.
Proof: Since T is a quasi-class A operator and it has a SVEP, then the result follows from [6] . For T∈B(H), it is known that the inclusion ea ea (f (T)) f ( (T)) σ ⊆ σ holds for every f Hol( (T)) ∈ σ , with no restriction on T. The next theorem shows that for quasi-class A operators the spectral mapping theorem holds for the essential approximate point spectrum. = Therefore ea f ( (T)). λ ∉ σ This completes the proof. Definition 8: [12] For T∈B(H) and closed subset F of the glocal spectral is about this subject the reader should refer to [11,12] .
Recall that generalized Weyl's theorem (g-Weyl's) holds for T if denotes the isolated points λ of σ(T), which are eigenvalues (no restriction on multiplicity) and is the set of all complex numbers λfor which T I − λ is not B-Weyl's. Berkani [3,proposition 3.2] has called an operator T∈B(H) is B-Fredholm if there exists a natural number n for which the induced operator n n n T : ran(T ) ran(T ) → is Fredholm in the usual sense and B-Weyl's" if in addition T n has zero index. Berkani [3,corollary 3.3] has shown that, if g-Weyl's theorem holds for T then so does Weyl's theorem.
For the sake of simplicity of notation we introduce the abbreviations gaW,aW,gW and W to signify that an operator T∈B(H) (which is usually understood) obeys generalized a-Weyl's theorem, a-Weyl's theorem, generalized Weyl's theorem and Weyl's theorem, respectively.
Analogous meaning is attached to the abbreviations gaB,aB,gB and B with respect to Browder's theorem. In the following diagrams, arrows signify implications between various Weyl's and Browder's theorems [2,4,5,20] . Proof: Since T is isoloid in (T) σ by [8,lemma 1.8] and has SVEP, then it suffices to prove that generalized Weyl's theorem holds for T. We shall show that

K(T I)
− λ is closed subspace [16,19] , Since , then by [16] there is some (T) λ ∉σ A bounded linear operator T is called a-isoloid if every isolated point of σ a (T) is an eigenvalue of T . Note that every a -isoloid operator is isoloid and the converse is not true in general. Theorem 2.4 of [21] affirms that if T * or T has the SVEP and if T is a-isoloid and generalized a-Weyl's holds for T then generalized a-Weyl's theorem holds for f(T), for every f Hol( (T)) ∈ σ . If T * is quasi-class A, then we have: Theorem 12: Let T * be a quasi-class A operator. Then generalized a-Weyl's theorem hold for T. and T or T* is a quasiclass A. Then the generalized a-Browder's theorem holds for T.

Proof:
The proof is a consequence immediate of [8,2] .

CONCLUSION
It can be shown that if T* is a quasi-class A then the generalized a-Browder's theorem holds for f(T) for every f Hol( (T)) ∈ σ .