On generalized Weyl’s type theorem

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN:2319-765X. Volume 10, Issue 2 Ver. IV (Mar-Apr. 2014), PP 35-41 www.iosrjournals.org On generalized Weyl’s type theorem 1,2 A. Babbah1, M. Zohry 2 (University Abdelmalek Essaadi, Faculty of Sciences, Mathematics Department,BP. 2121, Tetouan, Morocco ) Abstract: It is shown that if a bounded linear operator T or its adjoint T* has the single-valued extension property, then generalized Browder’s theorem holds for f(T) for every f ∈ H(σ(T)). We establish the spectral theorem for the B-Weyl spectrum which generalizes [15, Theorem 2.1] and we give necessary and sufficient conditions for such operator T to obey generalized Weyl’s theorem. Keywords: Single-valued extension property, Fredholm theory, generalized Weyl’s theorem, generalized Browder’s theorem. I.INTRODUCTION AND NOTATIONS Let X denote an infinite-dimensional complex Banach space and L( X ) the unital (with unit the identity operator, I , on X ) Banach algebra of bounded linear operators acting on X . For an operator T L(X) write T* for its adjoint, N(T) for its null space, R(T) for its range, σ(T)for its spectrum,  su  T  for its surjective spectrum,  a  T  for its approximate point spectrum,  (T ) for its nullity and  (T ) for its defect. T is called an upper semi-Fredholm (resp. a lower semi-Fredholm) operator if the range R(T) of T is closed and  (T ) < ∞ (resp.  (T ) < ∞). A semi-Fredholm operator is an upper or a lower semi-Fredholm operator. If both  (T ) and  (T ) are finite, then T is called a Fredholm operator and the index of T is defined by ind (T ) =  (T ) −  (T ) . For a T -invariant closed linear subspace restriction of T to Y. Y of X , let T / Y denote the operator given by the Tn to be the restriction of T to R(T n ) n n viewed as a map from R (T ) into itself. If for some integer n the range R (T ) is closed and Tn = T / R(Tn ) For a bounded linear operator T and for each integer n, define is a Fredholm (resp. semi-Fredholm) operator, then T is called a B-Fredholm (resp. semi-B-Fredholm) operator. Tm is a Fredholm operator and in dT ( m)in dT ( n)for each m  n . This permits to define the index of a B-Fredholm operator T as the index of the Fredholm operator Tn where, n n is any integer such that R (T ) is closed and Tn is a Fredholm operator. It is shown (see [2, Theorem 3.2]) that if S and T are two commuting B-Fredholm operators then the product ST is a B-Fredholm operator and i n d ( S T )  i n d () S  i n d () T . Let BF ( X ) be the class of all B-Fredholm operators and ( T )  {  ฀ : T  IB  F ( X ) } ( T ) ฀\ ( T )be be the B-Fredholm resolvent of T and let  B F B F B F the B-Fredholm spectrum of T. The class BF ( X ) has been studied by M. Berkani (see [3, Theorem 2.7]) where it was shown that an operator T L( X) is a B-Fredholm operator if and only if T S0 S1 where S0 is a Fredholm operator and S1 is a nilpotent one. He also proved that  BF (T ) is a closed subset of ฀ contained in the spectrum  (T ) and showed that the spectral mapping theorem holds for  BF (T ) , that is, f(  () T )  (() fT ) for any complex-valued analytic function on a neighborhood of  (T ) (see [3, B F B F Theorem 3.4]). From [21] we recall that for T L( X) , the ascent a (T ) and the descent d (T ) are given by In this case, from [3, Proposition 2.1]    n n  1    a ( T )i n f { n 0 :() N TN ( T ) } And n n  1 d ( T )i  n f {0 n  :() R TR  ( T ) } www.iosrjournals.org 35 | Page On generalized Weyl’s type theorem a (T ) and d (T ) are both finite the respectively, where the infinum over the emptyset is taken to be ∞. If aT ( )dT ( )p, X  N ( T) R ( T)and R(T ) is closed. p p p An operator T ∈ L(X) is called semi-regular if R(T) is a closed space and The semi-regular resolvent set is defined by NT ( )R (Tn) for every n ∈ N.   s  r e g T  {:  ฀ T  I i s s e m i  r e g u l a r }   we note s  r e g T  s  r e g  T is an open subset of ฀ . The semi-B-Fredholm resolvent set of T is given by *  ( T )  {:   ฀ T   I i s s e m i B F r e d h o l m } that . We recall that an operator T ∈ L(X) has the single-valued extension property, abbreviated SVEP, if, for S B F f :UX of the equation (T I)f()0for all  U is the zero function on U. We will denote by H((T)) the set of all complex-valued functions which are analytic on an open set containing  (T ) . As a consequence of [9, Théorème2.7], we obtain the following every open set U ⊆ C, the only analytic solution result. Proposition 1 Let T ∈ L(X). (i) If re g (T ). T  T has the SVEP then s a (ii) If T * has the SVEP then s re g (T ). T  s u For our investigations we need the following result. Proposition 2 Let T ∈ L(X). T has the SVEP then ind(T) 0 for every SBF(T). (ii) If T * has the SVEP then ind(T) 0for every SBF(T). (i) If SBF(T), then there exists an integer p such that the operator    Proof. (i) Let p p ( T / R ( T  I ) )I  ( T  I ) / R ( T  I ) is semi-Fredholm. From the Kato decomposition, there exists   0 such that p {  ฀ : 0  | |  }  s  r e g ( T / R ( T  I ) ) .      Since T has the single-valued extension property, Proposition 1 implies that   p p s r e g ( T / R ( T I ) ) = ( T / R ( T I ) ) . Therefore one verify that a   p p N ( ( T / R ( TI   ) )   I )  0 n d ( T  I )  i n d ( ( T / R ( T  I ) )  I )  0 and so i , 0  |    |   . holding for Thus, by the continuity of the index, ind(T) 0. (ii) This is included in part (i) since * in d (T )  in d (T ). T L(X) is said to be Weyl if it is Fredholm of index zero and Browder if it is Fredholm of finite ascent and descent. The essentiel spectrum  e (T ) , the Weyl spectrum  w (T ) and the Browder spectrum  b (T ) of T are defined by An operator  ( T )  {   ฀ : T   i s n o t F r e d h o l m } ;  ( T )  {   ฀ : T   i s n o t W e y l } ; e w  b (T )  {  ฀ : T   is not Browder} . It is well known that  ( T )   ( T )   ( T )   ( T ) . e w b An operator T L( X) is called B-Weyl if it is B-Fredholm of index zero. The B-Weyl spectrum T is defined by www.iosrjournals.org  BW (T ) of 36 | Page On generalized Weyl’s type theorem  ( T )  {   ฀ : T   i s n o t B W e y l } . B W For a subset K of C, we shall write iso(K) for its isolated points. A complex number 0 is said to be iso((T)) and the spectral projection corresponding to the set {0 } has T in L( X ) if  0 finite-dimensional range. The set of all Riesz points of T will be denoted by  0 (T ) . It is known that if T L(X) and  (T) then 0(T) if and only if T   I is Fredholm of finite ascent and descent () T  () T\ () T. (see [19]). Consequently  b 0 Let  (T ) denote the set of all poles of the resolvent of T and E0 (T ) denote the set {  ฀ : i s o ( ( T ) , 0  ( T  I )   } . For a normal operator T acting on a Hilbert space H, () T  () T\E () Twhere E (T ) is the set of all eigenvalues Berkani [2, Theorem 4.5] showed that  B W of T which are isolated in  (T ) . This result gives a generalization of the classical Weyl’s theorem  ( T )  ( T )\E ( T ). w 0 Riesz point of   II.SVEP AND GENERALIZED WEYL’S THEOREM The concept of Drazin invertibility plays an important role for the class of B-Fredholm operators. From [12] we recall that, for an algebra A with unit 1 we say that an element a ∈ A is Drazin invertible of degre k if there b aa , b a bb  ,a bb  a . The drazin spectrum of a ∈ A is defined by is an element b of A such that a  a  {   ฀ : a   1 i s n o t D r a z i n i n v e r t i b l e } .   k k In the case of A L(X) , it is well known that T is Drazin invertible if and only if it has a finite ascent and descent which is also equivalent to the fact that D T T0 T1 where T0 is an invertible operator and and [7, Corollary 2.2]. T1 is a nilpotent one, see for instance [12, Proposition 6]  ( T )   {  ( T  K ) : KX   () } where  ( X ) is the class of all compact operators ( T )   {  ( T  F ) : F  F ( X ) } . It was proved in [2, Theorem 4.3] that for T ∈ L(X),  Recall that acting on X . w B W Let T L(X) , we will say that : ( T )  ( T )\E ( T ). T satisfies Weyl’s theorem if  w 0 () T  () T\E () T. (ii) T satisfies generalized Weyl’s theorem if  B W () T  () T\ () T. (iii) T satisfies Browder’s theorem if  w 0 () T  () T\ () T. (iv) T satisfies generalized Browder’s theorem if  B W Recall from [5] that if T L( X) satisfies generalized Weyl’s theorem then it also satisfies Weyl’s (i) theorem and if T satisfies generalized Browder’s theorem then it satisfies Browder’s theorem. We now turn to an another extension of the characterization of operators obeying Weyl’s theorem ([1, Theorem4]). T L(X) then we have () T  () T\E () Tif and only if ET ( ) (T). (i)  B W Theorem 3 [4, Theorem 2.5] If (ii)  () T  () T\E () Tif and only if  (T) (T). B W B W D From this theorem we obtain immediately the following corollary. T L(X) , then T satisfies generalized Weyl’s theorem if () T  () T\ () Tand ET ( ) (T). and only if  B W Corollary 4 Let www.iosrjournals.org 37 | Page On generalized Weyl’s type theorem In [15, Theorem 2.1] it is proved that if either an operator T on an infinite dimensional separable Hilbert space or its Hilbert adjoint has the single-valued extension property, then the spectral mapping theorem holds for BWeyl spectrum. Using a standard argument and the Riesz functional calculus, we obtain the same result for operators on infinite dimensional Banach spaces with a simple and short proof. Proposition 5 Let Proof. Let (() fT )  f(  () T )for every f H((T)). T L(X) , then  B W B W  BW(f(T)), then f (T) I is not a B-Weyl’s operator. As (f( T ) )   ( f ( T ) )  f (  ( T ) ) , there exists  (T) such that   f () .    B W We have  m m m n 1 f ( z ) f ( ) = ( z ) ( z ) · · · ( z ) g ( z ) where g is a non vanishing analytic 1 n  m m m  (T ) . So f ( T ) f ( ) I ( T I ) ( T IT ) · · · ( I ) g ( T ) f ( T ) I   . 1 n Since f (T)- I is not a B-Weyl operator, and i n d ( f T  f ( ) I )  m  i n d ( T  I )  m  i n d () T  I      m  i n d () T  I   , 1 1 n n 1 function on there exists we get     {,  ,・ ・ ・ , }such that T   I 1 n BW(T). n is not a B-Weyl operator and since f ()   The opposite inclusion does not hold in general. Furthermore if f is injective on inclusion becomes an equality. The proof of the next result is similar to that one involving  w (T )  BW (T ) , the last (see [14, Theorem 3]). T L(X) , if f H((T)) is injective on  BW (T ) then  (() fT ) f(  () T ) . B W B W Theorem 6 Let BW ( X ) be the class of T L(X) such that in dT ( I)0for all BF(T) or in dT ( I)0for all BF(T) . We recall that hyponormals operators on a Hilbert space H lie in BW ( X ) . The following result shows that, for operators lying in the class BW ( X ) , the spectral mapping Let theorem for complex polynomials implies the spectral mapping one for complex-valued analytic functions. Theorem 7 For T L( X) verifying the single-valued extension property, the following assertions are equivalent : TBW(X).  () T )  (() fT )for all f H((T)). (ii) f( B W B W (i) (iii) p (  () T )  (() p T )for all complex polynomial p. _ B W B W Proof. (i)=⇒(ii) [22,Théorème 2.2.4] implies that for all f H((T)). (ii)=⇒(iii) Clear. f(  () T )  (() fT ) B W B W TBW(X). Then there are  ,  in  BF (T ) such that dT ( I)k and ind (T   I )  0 and ind (T   I )  0 . If we consider in l k  in dT (  I)l and the polynomial p ( t) ( t  ) ( t  ) , then p (T ) is (iii)=⇒(i) Assume that a B-Fredholm operator with BW(pT ( )). Since i n d (() p T )  lkkl  (  )  0 thus 0   BW(T) we get 0(  p )  p (B ( T ) )  ( p ( T ) ) a contradiction. W B W www.iosrjournals.org 38 | Page On generalized Weyl’s type theorem T or T* has the single-valued extension property, then f(  () T )  (() fT ) for any f H((T)). B W B W Proposition 8 If Proof. Let f H((T)). If T or T * has the SVEP, by Proposition 2, T lies in BW(X) and Theorem 7 concludes the proof. Let T L(X) , the analytical core of T is the subspace, K (T ) , defined below n x  X  x  c  x  x T x  x a n d x  c x f o r a l l n  { : ,0 : , 0 } .    n 01 n n n n The quasi-nilpotent part of T is the subspace 1 nn n   H T :  { x  X :l i m T x 0 } .   0 Both subspaces, will be of particular importance in what follows, they have been introduced and studied by Mbekhta (see [8–10]). In general neither H 0  T  nor K (T ) is closed. The following facts are easy to verify; T ( K ( TK ) )  ( T ) ,( K T )  H T   for every m  ฀ ; if x  X , then xH0 T if and only if 0 TxH0T. If T is invertible then H0T 0. m T L(X) , the following conditions are equivalent. (i)  is an isolated point of  (T ) .  H ( T   I )  K ( T I ) I) 0 and the direct (ii) X , where HT 0 0(  Theorem 9 [8, Theorem 1.6] Let sum is topological.  is a pole of the resolvent,  (T ) , of T of order p if and only if p p H ( T I ) N ( T I )      ( T   I ) R ( T   I ) and K . 0 Moreover, Our next goal is to show that generalized Browder’s theorem is satisfied for f  T  whenever T or T * has the single-valued extension property and f in H((T)) . The same result was showed in [6, Theorem 1.5] for the generalized a-Browder theorem. To settle our result, we use a characterization of the pole of the resolvent in terms of ascent and descent given in [13]. T verifie the single-valued extension property, then for any analytical function on an open neighbourhood of  (T ) , f (T ) verifie the single-valued extension property. Remark. It is shown in [18, Theorem 4.18] that if T L(X) or its adjoint has the single-valued extension property, then generalized Browder’s theorem holds for f (T ) for every f H((T)). Theorem 10 If   (T )\ (T ), so B W T   I is B-Weyl, hence B-Fredholm of index 0 and by [17, Theorem 1.82], T   I is Kato type. Since T or T * verifie SVEP, [17, Corollary 2.49] implies () T\ () T  () T. Conversely, ( T   I )  p ( T   I )   . Then  (T) and  that q B W Proof. Assume that if  (T) then  is isolated in  (T ) and by [4, Theorem 2.3], T   I is B-Weyl, () T\ () T  () T. Now if f H((T)), by the last remark and  B W f(T * )f(T )*  , f(T)orf(T*) verifie SVEP and consequently we obtain ( f ( T ) )  ( f ( T ) ) \  ( f ( T ) ) . B W  that is BW(T) and the fact that  From this theorem we obtain immediately the following corollary. Corollary 11 If T L(X) or its adjoint T * has the SVEP, then www.iosrjournals.org 39 | Page On generalized Weyl’s type theorem (T)ET ( ). T if and only if  (T* )E (T* ). (ii) Generalized-Weyl’s theorem holds for T * if and only if  (i) Generalized-Weyl’s theorem holds for The next result rewrite some results due to C. Schmoeger [13] as follows. Proposition 12 Let T ∈ L(X), the following conditions are equivalent  (T) . p ( T I ) N ( T I )      (ii)  E(T ) and there exists an integer p  1 for which H . 0 p ( T   I ) R ( T   I ). (iii)  E(T ) and there exists an integer p  1 for which K (iv)  E(T ) and T   I is of finite descent. (i) Proof. Without loss of generality we can assume that   0 . (i)⇒(ii) Since 0 is a pole of the resolvent of T of order p , it is an eigenvalue of T and (T)p . an isolated point of the spectrum of T . Hence 0E(T ) . Finally by Theorem 9 HT 0( )N N (Tp) and 0E(T ) from[8, Théorème 1.6] we have (ii)⇒(iii) If there exists p  1 such that HT 0( ) p p p X  H ( T ) K (T ). Then one obtain R ( T )  T ( H ( T ) )  T ( K ( T ) )and sinc HT N (Tp), 0 0 0( ) TK ( (T ))K (T )follows that KT ( )RT ( p). ( (T ))K (T )it follows that ( )RT ( p), since TK (iii)⇒(iv) If there exists p  1 such that KT p  1 p p R ()( T  T R ( T ) )(  T K ( T ) )  K ( T )( R T ) and d(T)  . (iv)⇒(i) Suppose that 0E(T ) and d(T)  . Since 0 is isolated in  (T ) , by [13, Theorem 4] X  H ( T ) K (T )and H0(T) 0is closed. Hence by [13, Theorem 2(b)] T has the SVEP at 0 0 and finally [13, Theorem 5] gives 0(T) . The following theorem follows immediately from Corollary 11 and Proposition 12. Theorem 13 Let T L( X) such that T or its adjoint T * has the single-valued extension property then the following conditions are equivalent: (i) Generalized Weyl’s theorem holds for T . 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