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Upper triangular operator matrices, asymptotic intertwining and Browder, Weyl theorems

Journal of Inequalities and Applications20132013:268

https://doi.org/10.1186/1029-242X-2013-268

  • Received: 8 February 2013
  • Accepted: 7 May 2013
  • Published:

Abstract

Given a Banach space X , let M C B ( X X ) denote the upper triangular operator matrix M C = ( A C 0 B ) , and let δ A B B ( B ( X ) ) denote the generalized derivation δ A B ( X ) = A X X B . If lim n δ A B n ( C ) 1 n = 0 , then σ x ( M C ) = σ x ( M 0 ) , where σ x stands for the spectrum or a distinguished part thereof (but not the point spectrum); furthermore, if R = R 1 R 2 B ( X X ) is a Riesz operator which commutes with M C , then σ x ( M C + R ) = σ x ( M C ) , where σ x stands for the Fredholm essential spectrum or a distinguished part thereof. These results are applied to prove the equivalence of Browder’s (a-Browder’s) theorem for M 0 , M C , M 0 + R and M C + R . Sufficient conditions for the equivalence of Weyl’s (a-Weyl’s) theorem are also considered.

MSC:47B40, 47A10, 47B47, 47A11.

Keywords

  • Banach space
  • asymptotically intertwined
  • SVEP
  • polaroid operator

1 Introduction

A Banach space operator T B ( X ) , the algebra of bounded linear transformations from a Banach space X into itself, satisfies Browder’s theorem if the Browder spectrum σ b ( T ) of T coincides with the Weyl spectrum σ w ( T ) of T; T satisfies Weyl’s theorem if the complement of σ w ( T ) in σ ( T ) is the set Π 0 ( T ) of finite multiplicity isolated eigenvalues of T. Weyl’s theorem implies Browder’s theorem, but the converse is generally false (see [13]). Let M 0 and M C B ( X X ) denote, respectively, the upper triangular operators M 0 = A B and M C = ( A C 0 B ) for some operators A , C , B B ( X ) . It is well known that σ x ( M 0 ) = σ x ( A ) σ x ( B ) = σ x ( M C ) { σ x ( A ) σ x ( B ) } for σ x = σ  or  σ b , and σ w ( M 0 ) σ w ( A ) σ w ( B ) = σ w ( M C ) { σ w ( A ) σ w ( B ) } . The problem of finding sufficient conditions ensuring the equality of the spectrum (and certain of its distinguished parts) of M 0 and M C , along with the problem of finding sufficient conditions for M 0 satisfies Browder’s theorem and/or Weyl’s theorem to imply M C satisfies Browder’s theorem and/or Weyl’s theorem (and vice versa), has been considered by a number of authors in the recent past (see [3], and some of the references cited there). For example, if either A or B has the single-valued extension property, SVEP for short, then σ ( M 0 ) = σ ( M C ) = σ ( A ) σ ( B ) . Again, if σ w ( M C ) = σ w ( A ) σ w ( B ) , then σ ( M 0 ) = σ ( M C ) = σ ( A ) σ ( B ) [[3], Proposition 3.2] and M 0 satisfies Browder’s theorem if and only if M C satisfies Browder’s theorem [[3], Theorem 4.8]; furthermore, in such a case, M 0 satisfies Weyl’s theorem if and only if M C satisfies Weyl’s theorem if and only if Π 0 ( M 0 ) = Π 0 ( M C ) [[3], Theorem 5.1]. The equality σ w ( M C ) = σ w ( A ) σ w ( B ) may be achieved in a number of ways: if either A and A , or A and B, or A and B , or B and B have SVEP, then σ w ( M C ) = σ w ( A ) σ w ( B ) [[3], Proposition 4.5]. In this paper we consider conditions of another kind, conditions which do not assume SVEP.

Given S , T B ( X ) , S and T are said to be asymptotically intertwined by X B ( X ) if lim n δ S T n ( X ) 1 n = 0 . Here δ S T B ( B ( X ) ) is the generalized derivation δ S T ( X ) = S X X T and δ S T n = δ S T ( δ S T n 1 ) . Evidently, S and T asymptotically intertwined by X does not imply T and S asymptotically intertwined by X. Furthermore, S and T asymptotically intertwined by X does not imply σ ( S ) = σ ( T ) , not even σ ( S ) σ ( T ) ; see [[4], Example 3.5.9]. However, as we shall see, if A, B, C are as in the definition of M C above, then A and B asymptotically intertwined by C implies the equality of the spectra, and many distinguished parts thereof to spectrum of M 0 and M C . We prove in the following that if lim n δ A B n ( C ) 1 n = 0 , then M C satisfies Browder’s theorem if and only if M 0 satisfies Browder’s theorem. If, additionally, the isolated points of σ ( M 0 ) are poles of the resolvent of M 0 , then M c satisfies Weyl’s theorem if and only if M 0 satisfies Weyl’s theorem. Extensions to a-Browder’s theorem, a-Weyl’s theorem and perturbations by Riesz operators are considered.

2 Notation and complementary results

For a bounded linear Banach space operator S B ( X ) , let σ ( S ) , σ p ( S ) , σ a ( S ) , σ s ( S ) and iso σ ( S ) denote, respectively, the spectrum, the point spectrum, the approximate point spectrum, the surjectivity spectrum and the isolated points of the spectrum of S. Let α ( S ) and β ( S ) denote the nullity and the deficiency of S, defined by
α ( S ) = dim S 1 ( 0 ) and β ( S ) = codim S ( X ) .
If the range S ( X ) of S is closed and α ( S ) < (resp. β ( S ) < ), then S is called an upper semi-Fredholm (resp. a lower semi-Fredholm) operator. If S B ( X ) is either upper or lower semi-Fredholm, S is called a semi-Fredholm operator, and ind ( S ) , the index of S, is then defined by ind ( S ) = α ( S ) β ( S ) . If both α ( S ) and β ( S ) are finite, then S is a Fredholm operator. The ascent, denoted asc ( S ) , and the descent, denoted dsc ( S ) , of S are given by
asc ( S ) = inf { n : S n ( 0 ) = S ( n + 1 ) ( 0 ) } , dsc ( S ) = inf { n : S n ( X ) = S n + 1 ( X ) }
(where the infimum is taken over the set of non-negative integers); if no such integer n exists, then asc ( S ) = , respectively dsc ( S ) = . Let
Φ + ( S ) = { λ C : S λ  is upper semi-Fredholm } , Φ ( S ) = { λ C : S λ  is lower semi-Fredholm } , Φ ( S ) = { λ C : S λ  is Fredholm } , σ S F + ( S ) = { λ σ a ( S ) : λ Φ + ( S ) } , σ S F ( S ) = { λ σ a ( S ) : λ Φ ( S ) } , σ e ( S ) = { λ σ ( S ) : λ Φ ( S ) } , σ w ( S ) = { λ σ ( S ) : λ σ e ( S )  or  ind ( S λ ) 0 } , σ a w ( S ) = { λ σ a ( S ) : λ σ S F + ( S )  or  ind ( S λ ) > 0 } , σ s w ( S ) = { λ σ s ( S ) : λ σ S F ( S )  or  ind ( S λ ) < 0 } , σ b ( S ) = { λ σ ( S ) : λ σ e ( S )  or  asc ( S λ ) dsc ( S λ ) } , σ a b ( S ) = { λ σ a ( S ) : λ σ S F + ( S )  or  asc ( S λ ) = } , σ s b ( S ) = { λ σ s ( S ) : λ σ S F ( S )  or  dsc ( S λ ) = } , Π 0 ( S ) = { λ iso σ ( S ) : 0 < dim ( S λ ) 1 ( 0 ) = α ( S λ ) < } , p 0 ( S ) = { λ iso σ ( S ) : λ Φ ( S ) , asc ( S λ ) = dsc ( S λ ) < } , H 0 ( S ) = { x X : lim n S n x 1 / n = 0 } .
Here σ w ( S ) is the Weyl spectrum, σ a w ( S ) denotes the Weyl (essential) approximate point spectrum, σ s w ( S ) the Weyl (essential) surjectivity spectrum, σ b ( S ) the Browder spectrum, σ a b ( S ) the Browder (essential) approximate point spectrum, σ s b ( S ) the Browder (essential) surjectivity spectrum, and H 0 ( S ) the quasi-nilpotent part of S [1]. Recall, [1], that H 0 ( S ) and K ( S ) , where K ( S ) denotes the analytic core
K ( S ) = { x X : there exists a sequence  { x n } X  and  δ > 0  for which x = x 0 , S ( x n + 1 ) = x n  and  x n δ n x  for all  n = 1 , 2 , } ,

are hyper-invariant (generally non-closed) subspaces of S such that S p ( 0 ) H 0 ( S ) for every integer p 0 and S K ( S ) = K ( S ) . Recall also that if 0 iso σ ( S ) , then X = H 0 ( S ) K ( S ) .

We say that S has the single valued extension property, or SVEP, at λ C if for every open neighborhood U of λ, the only analytic solution f to the equation ( S μ ) f ( μ ) = 0 for all μ U is the constant function f 0 ; we say that S has SVEP if S has a SVEP at every λ C . It is well known that finite ascent implies SVEP; also, an operator has SVEP at every isolated point of its spectrum (as well as at every isolated point of its approximate point spectrum).

S B ( X ) satisfies Browder’s theorem, shortened to S satisfies Bt, if σ w ( S ) = σ b ( S ) (if and only if σ ( S ) σ w ( S ) = p 0 ( S ) , see [[1], p.156]); S satisfies Weyl’s theorem, shortened to S satisfies Wt, if σ ( S ) σ w ( S ) = Π 0 ( S ) (if and only if S satisfies Bt and p 0 ( S ) = Π 0 ( S ) ) [[1], p.177]. The implication Wt Bt is well known.

An isolated point λ iso σ ( S ) is a pole (of the resolvent) of S B ( X ) if asc ( S λ ) = dsc ( S λ ) < . In such a case we say that S is polar at λ; we say that S is polaroid (resp., polaroid on a subset F of the set of isolated points of σ ( S ) ) if S is polar at every λ iso σ ( S ) (resp., at every λ F ). Let p ( S ) denote the set of poles of S.

Throughout the following, M 0 B ( X X ) shall denote the diagonal operator M 0 = A B and M C B ( X X ) shall denote the upper triangular operator matrix ( A C 0 B ) , for some operators A , B , C B ( X ) . Recall, [[5], Exercise 7, p.293], that asc ( A ) asc ( M C ) asc ( A ) + asc ( B ) and dsc ( B ) dsc ( M C ) dsc ( A ) + dsc ( B ) .

Lemma 2.1 If σ ( M 0 ) = σ ( M C ) , then p ( M 0 ) = p ( M C ) .

Proof Since σ ( M C ) = σ ( M 0 ) = σ ( A ) σ ( B ) , if a complex number λ p ( M C ) or p ( M 0 ) then λ iso ( σ ( A ) σ ( B ) ) . We consider the case in which λ iso σ ( A ) iso σ ( B ) : the argument works just as well for the case in which λ ρ ( A ) ( = C σ ( A ) ) or λ ρ ( B ) . Let λ p ( M C ) . Then
asc ( A λ ) asc ( M C λ ) < and dsc ( B λ ) dsc ( M C λ ) < .
If λ iso σ ( B ) and dsc ( B λ ) < , then asc ( B λ ) = dsc ( B λ ) < and B is polar at λ [[1], Theorem 3.81]. Now let λ iso σ ( A ) . Since M C is polar at λ, H 0 ( M C λ ) = ( M C λ ) p ( 0 ) for some integer p 1 . Observe that
H 0 ( A λ ) = H 0 ( M C λ ) X = ( M C λ ) p ( 0 ) X = ( A λ ) p ( 0 ) .
Hence, if λ iso σ ( A ) , then
X = H 0 ( A λ ) K ( A λ ) = ( A λ ) p ( 0 ) K ( A λ ) ( A λ ) p X = 0 ( A λ ) p K ( A λ ) = K ( A λ ) X = ( A λ ) p ( 0 ) ( A λ ) p X ,
i.e., A is polar at λ. Now, since
asc ( M 0 λ ) asc ( A λ ) + asc ( B λ ) and dsc ( M 0 λ ) dsc ( A λ ) + dsc ( B λ ) ,
we have
asc ( M 0 λ ) = dsc ( M 0 λ ) < ,

i.e., M 0 is polar at λ. Conversely, if λ p ( M 0 ) , then asc ( M 0 λ ) = max { asc ( A λ ) , asc ( B λ ) } and dsc ( M 0 λ ) = max { dsc ( A λ ) , dsc ( B λ ) } implies asc ( M C λ ) asc ( A λ ) + asc ( B λ ) and dsc ( M C λ ) dsc ( A λ ) + dsc ( B λ ) are both finite, hence equal. Thus M C is polar at λ. □

Remark 2.2 A number of conditions guaranteeing (the spectral equality) σ ( M C ) = σ ( M 0 ) are to be found in the literature. Thus, for example, if A or B has SVEP, or if σ w ( M C ) = σ w ( A ) σ w ( B ) , or σ a w ( M C ) = σ a w ( A ) σ a w ( B ) [[3], (I) p.5 and Proposition 3.2], then σ ( M C ) = σ ( M 0 ) . Compact operators have SVEP; hence, if either of A or B is compact, then σ ( M C ) = σ ( M 0 ) .

Lemma 2.1 shows that if B is a compact operator then p ( M 0 ) = p ( M C ) . A proof of the following lemma may be obtained from that of Lemma 2.1: we give here an independent proof, exploiting the additional information contained in the hypothesis.

Lemma 2.3 If σ ( M 0 ) = σ ( M C ) , then p 0 ( M 0 ) = p 0 ( M c ) .

Proof Once again we consider points λ iso σ ( A ) iso σ ( B ) . Let λ p 0 ( M C ) . Then α ( M C λ ) = β ( M C λ ) < implies M C λ Φ , and this in turn implies A λ Φ + and B λ Φ . Since λ is isolated in σ ( A ) and σ ( B ) , λ p 0 ( A ) p 0 ( B ) [[1], Theorem 3.77]. Consequently, λ p ( M 0 ) ; furthermore, since α ( M 0 λ ) α ( A λ ) + α ( B λ ) , λ p 0 ( M 0 ) . Conversely, if λ p 0 ( M 0 ) , then A λ and B λ Φ , and hence (since λ is isolated in σ ( A ) and σ ( B ) ) λ p 0 ( A ) p 0 ( B ) . This, as above, implies λ p 0 ( M C ) . □

The following technical lemma will be required in the sequel.

Lemma 2.4 If A is polaroid on Π 0 ( M C ) and σ ( M C ) = σ ( M 0 ) , then Π 0 ( M C ) Π 0 ( M 0 ) .

Proof Evidently, ( M C λ ) 1 ( 0 ) implies ( M 0 λ ) 1 ( 0 ) , and α ( M C λ ) < implies α ( A λ ) < . Let λ Π 0 ( M C ) ; then λ iso σ ( M 0 ) . We prove that α ( B λ ) < . Suppose to the contrary that α ( B λ ) = . Since
( M C λ ) ( x y ) = { ( A λ ) x + C y } ( B λ ) y ,

either dim ( C ( B λ ) 1 ( 0 ) ) < or dim ( C ( B λ ) 1 ( 0 ) ) = . If dim ( C ( B λ ) 1 ( 0 ) ) < , then (since α ( B λ ) = ) ( B λ ) 1 ( 0 ) contains an orthonormal sequence { y j } such that ( M C λ ) ( 0 y j ) = 0 for all j = 1 , 2 ,  . But then α ( M C λ ) = , a contradiction. Hence dim ( C ( B λ ) 1 ( 0 ) ) = . Since λ ρ ( A ) iso σ ( A ) and A is (by hypothesis) polar at λ (with, as observed above, α ( A λ ) < ) α ( A λ ) = β ( A λ ) < . Thus dim { C ( B λ ) 1 ( 0 ) ( A λ ) X } = , and so there exists a sequence { x j } such that ( A λ ) x j = C y j for all j = 1 , 2 ,  . But then ( M C λ ) ( x j y j ) = 0 for all j = 1 , 2 ,  , and hence α ( M C λ ) = . This contradiction implies that we must have α ( B λ ) < . Since α ( M 0 λ ) α ( A λ ) + α ( B λ ) , we conclude that λ Π 0 ( M 0 ) . □

Let δ S T B ( B ( X ) ) denote the generalized derivation δ S T ( X ) = S X X T , and define δ S T n by δ S T n 1 ( δ S T ) . The operators S , T B ( X ) are said to be asymptotically intertwined by the identity operator I B ( X ) if lim n δ S T n ( I ) 1 n = 0 ; S, T are said to be quasi-nilpotent equivalent if lim n δ S T n ( I ) 1 n = lim n δ T S n ( I ) 1 n = 0 [[4], p.253]. Quasi-nilpotent equivalence preserves a number of spectral properties [[4], Proposition 3.4.11]. In particular:

Lemma 2.5 Quasi-nilpotent equivalent operators have the same spectrum, the same approximate point spectrum and the same surjectivity spectrum.

3 Results

Let K ( X ) denote the ideal of compact operators in B ( X ) . The following construction, known in the literature as the Sadovskii/Buoni, Harte and Wickstead construction [[6], p.159], leads to a representation of the Calkin algebra B ( X ) / K ( X ) as an algebra of operators on a suitable Banach space. Let S B ( X ) . Let ( X ) denote the Banach space of all bounded sequences x = ( x n ) n = 1 of elements of X endowed with the norm x : = sup n N x n , and write S , S x : = ( S x n ) n = 1 for all x = ( x n ) n = 1 , for the operator induced by S on ( X ) . The set m ( X ) of all precompact sequences of elements of X is a closed subspace of ( X ) which is invariant for S . Let X q : = ( X ) / m ( X ) , and denote by S q the operator S on X q . The mapping S S q is then a unital homomorphism from B ( X ) B ( X q ) with kernel K ( X ) which induces a norm decreasing monomorphism from B ( X ) / K ( X ) to B ( X q ) with the following properties (see [[6], Section 17] for details):
  1. (i)

    S is upper semi-Fredholm, S Φ + , if and only if S q is injective, if and only if S q is bounded below;

     
  2. (ii)

    S is lower semi-Fredholm, S Φ , if and only if S q is surjective;

     
  3. (iii)

    S is Fredholm, S Φ , if and only if S q is invertible.

     

Lemma 3.1 For every S B ( X ) , σ e ( S ) = σ ( S q ) , σ S F + ( S ) = σ a ( S q ) and σ S F ( S ) = σ s ( S q ) .

Proof The following implications hold:
λ σ S F + ( S ) S λ Φ + ( S λ ) q  is bounded below λ σ S F + ( S ) λ σ a ( S q ) , λ σ S F ( S ) S λ Φ ( S λ ) q  is onto λ σ S F ( S ) λ σ s ( S q ) and λ σ e ( S ) S λ Φ ( S λ ) q  is invertible λ σ ( S q ) .

 □

The following theorem is essentially known [7] we provide here an alternative proof, using quasi-nilpotent equivalence and the construction above. Let Σ 0 denote either of σ e , σ S F + , σ S F , σ w , σ a w , σ s w , σ b , σ a b and σ s b .

Theorem 3.2 Let S , R B ( X ) . If R is a Riesz operator which commutes with S, then σ x ( S + R ) = σ x ( S ) , where σ x Σ 0 .

Proof It is clear from the definition of a Riesz operator R B ( X ) that R μ is Browder (i.e., μ σ b ( R ) ), and a-Browder and s-Browder, for all non-zero μ σ ( R ) (see, for example, [[1], Theorem 3.111]). Hence σ ( R q ) = { 0 } , i.e., R q B ( X q ) is quasi-nilpotent. Let t [ 0 , 1 ] ; then S commutes with tR and ( S + t R ) q = S q + t R q . It follows that
lim n δ ( S + t R ) q S q n ( I q ) 1 n = lim n δ S q ( S + t R ) q n ( I q ) 1 n = 0 ,
i.e., S q and S q + t R q are quasi-nilpotent equivalent operators for all t [ 0 , 1 ] . Thus σ x ( ( S + R ) q ) = σ x ( S q ) , where σ x = σ  or  σ a  or  σ s . Hence
σ x ( S + R ) = σ x ( S ) ; σ x = σ e  or  σ a e  or  σ s e .
The semi-Fredholm index being a continuous function, we also have from the above that
σ x ( S + R ) = σ x ( S ) ; σ x = σ w  or  σ a w  or  σ s w .

To complete the proof, we prove next that σ b ( S + R ) = σ b ( S ) ; the proof for σ a b and σ s b is similar, and left to the reader. It would suffice to prove that 0 σ b ( S ) 0 σ b ( S + R ) . Suppose that 0 σ b ( S ) . Then S Φ (and asc ( S ) = dsc ( S ) < ), hence S + t R Φ for all t [ 0 , 1 ] . For an operator T, let N ( T ) ¯ and T ( X ) denote, respectively, the closure of the hyper kernel and the hyper range of T. Then N ( S + t R ) ¯ ( S + t R ) ( X ) is constant on [ 0 , 1 ] , and so, since N ( S ) ¯ S ( X ) = N ( S ) S ( X ) = { 0 } , it follows that N ( S + R ) ( S + R ) ( X ) = { 0 } . Consequently, S + R has SVEP at 0 [[1], Corollary 2.26]. But then since S + R Φ , S + R is Browder. Considering S = ( S + R ) R proves 0 σ b ( S + R ) 0 σ b ( S ) . □

The following lemma appears in [[8], Lemma 2.3]. Let Π 0 f ( S ) = { λ iso σ ( S ) : α ( S λ ) < } . Clearly, Π 0 ( S ) Π 0 f ( S ) .

Lemma 3.3 If S , R B ( X ) , and R is a Riesz operator which commutes with S, then Π 0 f ( S + R ) σ ( S ) iso σ ( S ) .

Let Σ = Σ 0 σ σ a σ s .

Theorem 3.4 If lim n δ A B n ( C ) 1 n = 0 , then σ x ( M C ) = σ x ( M 0 ) , where σ x Σ .

Proof A straightforward calculation shows that
δ M C M 0 n ( I ) = δ M 0 M C n ( I ) = ( 0 δ A B n 1 ( C ) 0 0 ) .
Hence
lim n δ M C M 0 n ( I ) 1 n = lim n δ M 0 M C n ( I ) 1 n lim n δ A B n 1 ( C ) 1 n = 0 ,
i.e., M C and M 0 are quasi-nilpotent equivalent. Similarly, writing M C ( q ) for ( M C ) q and M 0 ( q ) for ( M 0 ) q ,
lim n δ M C ( q ) M 0 ( q ) n ( I q ) 1 n = lim n δ M 0 ( q ) M C ( q ) n ( I q ) 1 n lim n δ A q B q n 1 ( C q ) 1 n = lim n δ A B n 1 ( C ) 1 n = 0 ,
i.e., M C ( q ) and M 0 ( q ) are quasi-nilpotent equivalent (in B ( ( X X ) q ) ). Hence σ x ( M C ) = σ x ( M 0 ) , where σ x = σ  or  σ a  or  σ s  or  σ e  or  σ S F +  or  σ S F . Since
M 0 = ( A 0 0 I ) ( I 0 0 B ) = ( I 0 0 B ) ( A 0 0 I )
and
M C = ( I 0 0 B ) ( I C 0 I ) ( A 0 0 I ) ,
where ( I C 0 I ) is invertible, and since λ σ e ( M C ) λ σ e ( M 0 ) A λ , B λ Φ (similarly, λ σ S F + ( M C ) A λ , B λ Φ + and λ σ S F ( M C ) A λ , B λ Φ ), ind ( M C λ ) = ind ( A λ ) + ind ( B λ ) = ind ( M 0 λ ) . Hence σ x ( M C ) = σ x ( M 0 ) , where σ x = σ w  or  σ a w  or  σ s w . Observe that
σ b ( M C ) = { λ σ ( M C ) : λ σ w ( M C )  or  λ iso σ ( M C ) } σ b ( M C ) = { λ σ ( M 0 ) : λ σ w ( M 0 )  or  λ iso σ ( M 0 ) } , σ a b ( M C ) = { λ σ a ( M C ) : λ σ a w ( M C )  or  λ iso σ a ( M C ) } σ a b ( M C ) = { λ σ a ( M 0 ) : λ σ a w ( M 0 )  or  λ iso σ a ( M 0 ) }
and
σ s b ( M C ) = { λ σ s ( M C ) : λ σ s w ( M C )  or  λ iso σ s ( M C ) } σ s b ( M C ) = { λ σ s ( M 0 ) : λ σ s w ( M 0 )  or  λ iso σ s ( M 0 ) }

[[1], Corollary 3.23, Theorem 3.23 and Theorem 3.27]. Hence σ x ( M C ) = σ x ( M 0 ) , where σ x = σ b  or  σ a b  or  σ s b . □

Remark 3.5 If M B ( X X ) is the operator M = ( A C D B ) such that the entries A, B, C and D mutually commute, then σ x ( M ) = { λ C : 0 σ x ( ( A λ ) ( B λ ) C D ) } [[9], Theorem 2.3], where σ x = σ  or  σ e . Dispensing with the mutual commutativity hypothesis and assuming instead that C D = D C = 0 , C commutes with A and B, and lim n δ A B n ( D ) 1 n = 0 , an argument similar to that used to prove Theorem 3.4 shows that σ x ( M ) = σ x ( M C ) , where σ x = σ  or  σ a  or  σ s  or  σ e  or  σ S F ± .

Theorem 3.6 Suppose that lim n δ A B n ( C ) 1 n = 0 . Then:
  1. (a)

    M C satisfies Bt if and only if M 0 satisfies Bt.

     
  2. (b)

    Let R i B ( X ) , i = 1 , 2 , be Riesz operators such that R = R 1 R 2 commutes with M C . Then M 0 satisfies Bt M C + R satisfies Bt M 0 + R satisfies Bt M C satisfies Bt.

     
Proof The hypothesis R commutes with M C implies R commutes with M 0 , R 1 C = C R 2 and δ ( M C + R ) ( M 0 + R ) n ( I ) = δ M C M n ( I ) .
  1. (a)
    Recall that an operator S satisfies Bt if and only if σ w ( S ) = σ b ( S ) . Hence the following implications hold:
    M 0  satisfies Bt σ w ( M 0 ) = σ b ( M 0 ) σ w ( M c ) = σ b ( M C ) (Theorem 3.4) M C  satisfies Bt .
     
  2. (b)
    The hypothesis lim n δ A B n ( C ) 1 n = 0 implies that M C + R and M 0 + R are quasi-nilpotent equivalent ( by Theorem 3.4 that σ x ( M C + R ) = σ x ( M 0 + R ) , where σ x Σ ). The operator R being Riesz, Theorem 3.2 implies σ x ( T + R ) = σ x ( T ) , where T = M C or M 0 and σ x = σ w or σ b . The (two way) implications
    M 0  satisfies Bt σ w ( M 0 ) = σ b ( M 0 ) σ w ( M 0 + R ) = σ b ( M 0 + R ) ( M 0 + R  satisfies Bt ) σ w ( M C + R ) = σ b ( M C + R ) M C + R  satisfies Bt σ w ( M C ) = σ b ( M C ) M C  satisfies Bt
     

now complete the proof. □

Remark 3.7 (i) S B ( X ) satisfies a-Browder’s theorem, a-Bt, if and only if σ a w ( S ) = σ a b ( S ) (equivalently, if and only if σ a ( S ) σ a w ( S ) = p 0 a ( S ) = { λ iso σ a ( S ) : S λ Φ + } = { λ σ a ( S ) : S λ Φ + , asc ( S λ ) < } [[2], Theorem 3.3]). Theorem 3.6 holds with Bt replaced by a-Bt. (Thus, if either M 0 or M C satisfies a-Bt, then M 0 , M C , M 0 + R and M C + R all satisfy a-Bt.) Furthermore, since S satisfies generalized Browder’s theorem, gBt, if and only if it satisfies Bt and S satisfies generalized a-Browder’s theorem, a-gBt, if and only if it satisfies a-Bt [10], Bt may be replaced by gBt or a-gBt in Theorem 3.6. Here, we refer the interested reader to consult [2, 10] for information about gBt and a-gBt.
  1. (ii)

    The equivalence S satisfies Bt S satisfies Bt is well known. This does not hold for a-Bt: S satisfies a-Bt does not imply S satisfies a-Bt (or vice versa). We say that S satisfies s-Bt if S satisfies a-Bt (equivalently, if σ s b ( S ) = σ s w ( S ) ). It is easily seen, we leave the verification to the reader, if either M 0 or M C satisfies s-Bt, then (in Theorem 3.6) M 0 , M C , M 0 + R and M C + R all satisfy s-Bt.

     

We consider next a sufficient condition for the equivalence of Weyl’s theorem for operators M 0 and M C such that lim n δ A B n ( C ) 1 n = 0 . We say in the following that an operator S is finitely polaroid on a subset F iso σ ( S ) if every λ F is a finite rank pole of S. Evidently, M 0 is finitely polaroid if and only if A and B are finitely polaroid.

Theorem 3.8 Suppose that lim n δ A B n ( C ) 1 n = 0 .
  1. (a)

    If A is polaroid, then M C satisfies Wt if and only if M 0 satisfies Wt.

     
  2. (b)

    Let R i B ( X ) , i = 1 , 2 , be Riesz operators such that R = R 1 R 2 commutes with M C . A sufficient condition for the equivalence M C + R satisfies Wt M 0 + R satisfies Wt is that M 0 is finitely polaroid.

     
Proof (a) If M C satisfies Wt, then σ ( M C ) σ w ( M C ) = p 0 ( M C ) = Π 0 ( M C ) . Since σ ( M 0 ) = σ ( M C ) and σ w ( M C ) = σ w ( M 0 ) (Theorem 3.4) and since Wt implies Bt, Theorem 3.6(a) implies σ ( M 0 ) σ w ( M 0 ) = p 0 ( M 0 ) Π 0 ( M 0 ) . Consequently, Π 0 ( M C ) Π 0 ( M 0 ) . Let λ Π 0 ( M 0 ) . Then λ iso σ ( M C ) , α ( A λ ) < and α ( B λ ) < . Hence, since α ( A λ ) α ( M C λ ) α ( A λ ) + α ( B λ ) , α ( M C λ ) < . Evidently, λ iso σ ( A ) ρ ( A ) . If λ iso σ ( A ) , then A polaroid implies 0 < α ( A λ ) , and hence 0 < α ( M C λ ) . If instead λ ρ ( A ) , then ( A λ ) 1 C x x ( M C λ ) 1 ( 0 ) for every x ( B λ ) 1 ( 0 ) ; once again, 0 < α ( M C λ ) . Consequently, λ Π 0 ( M C λ ) = p 0 ( M C λ ) = p 0 ( M 0 λ ) and hence Π 0 ( M 0 ) = p 0 ( M 0 ) M 0 satisfies Wt. Conversely, if M 0 satisfies Wt, then σ ( M C ) σ w ( M C ) = p 0 ( M C ) = p 0 ( M 0 ) = Π 0 ( M 0 ) = σ ( M 0 ) σ w ( M 0 ) and Π 0 ( M 0 ) Π 0 ( M C ) . Since A is polaroid (hence polar on Π 0 ( M C ) ) and σ ( M 0 ) = σ ( M C ) , Lemma 2.4 implies Π 0 ( M 0 ) = Π 0 ( M C ) . Thus M C satisfies Wt.
  1. (b)
    Start by observing that σ ( M 0 ) = σ ( M C ) , and hence M C is finitely polaroid if and only if M 0 is finitely polaroid (Lemma 2.3). Suppose M 0 + R satisfies Wt. Then the implication Wt Bt combined with Theorem 3.6(b) implies that both M 0 + R and M C + R satisfy Bt. As noted in the proof of Theorem 3.6(b), σ w ( T + R ) = σ w ( T ) , T = M 0 or M C . Furthermore, since M 0 + R and M C + R are quasi-nilpotent equivalent, σ x ( M 0 + R ) = σ x ( M C + R ) , σ x = σ  or  σ w (Theorem 3.4). Hence
    Π 0 ( M 0 + R ) = σ ( M 0 + R ) σ w ( M 0 + R ) = σ ( M C + R ) σ w ( M C + R ) = p 0 ( M C + R ) Π 0 ( M C + R ) .
     

If λ Π 0 ( M C + R ) and λ σ ( M C ) , then ( M C λ ) is invertible and so M C λ Φ M C + R λ Φ . Hence, since λ iso σ ( M C + R ) , λ p 0 ( M C + R ) . If, instead, λ σ ( M C ) , then λ iso σ ( M C ) (Lemma 3.3) λ iso σ ( M 0 ) λ p 0 ( M 0 ) (since M 0 is finitely polaroid) λ p 0 ( M C ) (Lemma 2.3) M C λ Φ , and this as above implies λ p 0 ( M c + R ) . Hence Π 0 ( M C + R ) = p 0 ( M C + R ) , and M C + R satisfies Wt. The converse, M C + R satisfies Wt M 0 + R satisfies Wt follows from a similar argument (recall that M C is finitely polaroid follows from the hypothesis that M 0 is finitely polaroid). □

Remark 3.9 The equivalence of Theorem 3.8(b) extends to
M 0  satisfies Bt M 0 + R  satisfies Wt M C + R  satisfies Wt M C  satisfies Bt .

This is seen as follows. The implication M 0 + R satisfies Wt M 0 satisfies Bt and M C + R satisfies Wt M C satisfies Bt are clear from Theorem 3.6(b). If M 0 satisfies Bt, then the hypothesis M 0 is finitely polaroid implies M 0 satisfies Wt. By Theorem 3.6(b), M 0 + R satisfies Bt, i.e., σ ( M 0 + R ) σ w ( M 0 + R ) = p 0 ( M 0 + R ) Π 0 ( M 0 + R ) . Let λ Π 0 ( M 0 + R ) . If λ σ ( M 0 ) , then ( M 0 λ Φ ) M 0 + R λ Φ λ p 0 ( M 0 + R ) (since λ iso σ ( M 0 + R ) ); if λ σ ( M 0 ) , then λ iso σ ( M 0 ) (by Lemma 3.3) and so (since M 0 is finitely polaroid) λ p 0 ( M 0 ) M 0 λ Φ M 0 + R λ Φ λ p 0 ( M 0 + R ) . Thus, in either case, Π 0 ( M 0 + R ) p 0 ( M 0 + R ) , and hence M 0 + R satisfies Wt. The proof for M C satisfies Bt M C + R satisfies Wt is similar: recall from Lemma 2.3 that M 0 finitely polaroid implies M C finitely polaroid.

a-Wt. T B ( X ) satisfies a-Weyl’s theorem, a-Wt for short, if T satisfies a-Bt and p 0 a ( T ) = Π 0 a ( T ) (equivalently, if σ a ( T ) σ a w ( T ) = p 0 a ( T ) = Π 0 a ( T ) ), where Π 0 a ( T ) = { λ iso σ a ( T ) : 0 < α ( T λ ) < } [1]. We say in the following that T is a-polaroid if T is polar at every λ iso σ a ( T ) . Trivially, a-polaroid implies polaroid (indeed, p 0 a ( T ) = p 0 ( T ) in such a case), but the converse is not true in general. Theorem 3.8 has an a-Wt analogue, which we prove below. We note, however, that the perturbation of an operator by a commuting Riesz operator preserves neither its spectrum nor its approximate point spectrum: this will, per se, force us into making an assumption on the approximate point spectrum of M 0 and M 0 + R in the analogue of Theorem 3.8(b).

Theorem 3.10 Suppose that lim n δ A B n ( C ) 1 n = 0 .
  1. (a)

    If M 0 is a-polaroid, then M C satisfies a-Wt if and only if M 0 satisfies a-Wt.

     
  2. (b)

    Let R i B ( X ) , i = 1 , 2 , be Riesz operators such that R = R 1 R 2 commutes with M C . If σ a ( M 0 ) = σ a ( M 0 + R ) , then a sufficient condition for the equivalence M C + R satisfies a - Wt M 0 + R satisfies a-Wt is that M 0 is finitely a-polaroid.

     
Proof (a) We prove Π 0 a ( M 0 ) = Π 0 a ( M C ) : the proof of (a) would then follow from the fact that if M 0 satisfies a-Wt ( M 0 satisfies a -Bt M C satisfies a-Bt), then
Π 0 a ( M 0 ) = σ a ( M 0 ) σ a w ( M 0 ) = σ a ( M C ) σ a w ( M C ) = p 0 a ( M C ) Π 0 a ( M C )
and if M C satisfies a-Wt, then
Π 0 a ( M C ) = σ a ( M C ) σ a w ( M C ) = σ a ( M 0 ) σ a w ( M 0 ) = p 0 a ( M 0 ) Π 0 a ( M 0 ) .
If λ Π 0 a ( M 0 ) , then
λ iso σ a ( M 0 ) , 0 < α ( M 0 λ ) < λ p 0 ( M 0 ) ( since  M 0  is  a -polaroid ) λ ( p 0 ( A ) p 0 ( B ) ) ( p 0 ( A ) ρ ( B ) ) ( ρ ( A ) p 0 ( B ) ) α ( M C λ ) α ( A λ ) + α ( B λ ) < , asc ( M C λ ) asc ( A λ ) + asc ( B λ ) < , dsc ( M C λ ) dsc ( A λ ) + dsc ( B λ ) < asc ( M C λ ) = dsc ( M C λ ) < , 0 < α ( M C λ ) < λ p 0 ( M C ) Π 0 ( M C ) Π 0 a ( M C ) ;
if instead λ Π 0 a ( M C ) , then
λ iso σ a ( M C ) , 0 < α ( M C λ ) < λ iso σ a ( M 0 ) , 0 < α ( M C λ ) < λ p ( M 0 ) , 0 < α ( M C λ ) < λ p 0 ( M c ) (Lemma 2.4) λ p 0 ( M 0 ) (Lemma 2.1) λ Π 0 ( M 0 ) Π 0 a ( M C ) .
  1. (b)
    If σ a ( M 0 + R ) = σ a ( M 0 ) , then it follows from Lemma 2.4 and Theorem 3.4 that
    σ x ( M 0 ) = σ x ( M 0 + R ) = σ x ( M C + R ) = σ x ( M C ) ; σ x = σ a  or  σ a w .
     
Recall from Remark 3.7 that if either of M 0 + R or M C + R satisfies a-Bt, then M 0 , M 0 + R , M C and M C + R all satisfy a-Bt. Hence, in view of the spectral equalities above,
p 0 a ( M 0 ) = p 0 a ( M C ) = p 0 a ( M C + R ) = p 0 a ( M 0 + R ) ,

whenever either of M 0 , M 0 + R , M C and M C + R satisfies a-Bt. Observe that the hypothesis M 0 is finitely a-polaroid implies p 0 a ( M 0 ) = p 0 ( M 0 ) = p 0 ( M C ) = p 0 a ( M 0 + R ) ; hence (since p 0 a ( M 0 ) = p 0 a ( M C ) = p 0 a ( M C + R ) = p 0 a ( M 0 + R ) ) p 0 a ( S ) = p 0 a ( T ) for every choice of S , T = M 0  or  M C  or  M 0 + R  or  M C + R . We prove now that if either of M 0 + R and M C + R satisfies a-Wt, then Π 0 a ( M 0 + R ) = Π 0 a ( M C + R ) : this would then imply that if one satisfies a-Wt, then so does the other.

Suppose M 0 + R satisfies a-Wt. Then p 0 ( M 0 + R ) = p 0 a ( M 0 + R ) = Π 0 a ( M 0 + R ) ( Π 0 a ( M 0 + R ) = Π 0 ( M 0 + R ) ) and Π 0 a ( M 0 + R ) Π 0 a ( M C + R ) . Let λ Π 0 a ( M c + R ) ; then λ iso σ a ( M C + R ) = iso σ a ( M 0 ) implies λ p 0 ( M 0 ) = p 0 a ( M C + R ) . Thus Π 0 a ( M C + R ) p 0 a ( M C + R ) = p 0 a ( M 0 + R ) = Π 0 a ( M 0 + R ) . Consequently, Π 0 a ( M 0 + R ) = Π 0 a ( M C + R ) in this case. Suppose next that M C + R satisfies a-Wt. Then p 0 ( M C + R ) = p 0 a ( M C + R ) = Π 0 a ( M C + R ) and Π 0 a ( M C + R ) Π 0 a ( M 0 + R ) . Let λ Π 0 a ( M 0 + R ) ; then λ iso σ a ( M 0 ) implies λ p 0 a ( M 0 ) = p 0 a ( M C + R ) . As above, this implies Π 0 a ( M 0 + R ) = Π 0 a ( M C + R ) . □

The following corollary is immediate from Theorem 3.10(b).

Corollary 3.11 Suppose that lim n δ A B n ( C ) 1 n = 0 . If R i B ( X ) , i = 1 , 2 , are quasi-nilpotent operators such that R = R 1 R 2 commutes with M C , then a sufficient condition for the equivalence M C + R satisfies a - Wt M 0 + R satisfies a-Wt is that M 0 is finitely a-polaroid.

Authors’ information

Work carried out together whilst the first author was visiting Korea.

Declarations

Acknowledgements

This work was supported by the Incheon National University Research Grant in 2012.

Authors’ Affiliations

(1)
8 Redwood Grove, Northfield Avenue, Ealing, London, W5 4SZ, United Kingdom
(2)
Department of Mathematics Education, Seoul National University of Education, Seoul, 137-742, Korea
(3)
Department of Mathematics, Incheon National University, Incheon, 406-772, Korea

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