Open Access

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: 29 May 2013

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)
(2)
Department of Mathematics Education, Seoul National University of Education
(3)
Department of Mathematics, Incheon National University

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