Abstract
AbstractGiven a Banach spaceX, letMC∈B(X⊕X)denote the upper triangular operator matrixMC=(AC0B), and letδAB∈B(B(X))denote the generalized derivationδAB(X)=AX−XB. Iflimn→∞∥δABn(C)∥1n=0, thenσx(MC)=σx(M0), whereσxstands for the spectrum or a distinguished part thereof (but not the point spectrum); furthermore, ifR=R1⊕R2∈B(X⊕X)is a Riesz operator which commutes withMC, thenσx(MC+R)=σx(MC), whereσxstands 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 forM0,MC,M0+RandMC+R. Sufficient conditions for the equivalence of Weyl’s (a-Weyl’s) theorem are also considered.MSC:47B40, 47A10, 47B47, 47A11.
Highlights
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 σb(T)of T coincides with the Weyl spectrum σw(T) of T; T satisfies Weyl’s theorem if the complement of σw(T) in σ (T) is the set (T) of finite multiplicity isolated eigenvalues of T
It is well known that σx(M ) = σx(A) ∪ σx(B) = σx(MC) ∪ {σx(A) ∩ σx(B)} for σx = σ or σb, and σw(M ) ⊆
The problem of finding sufficient conditions ensuring the equality of the spectrum of M and MC(q) for (MC), along with the problem of finding sufficient conditions for M satisfies Browder’s theorem and/or Weyl’s theorem to imply MC satisfies Browder’s theorem and/or Weyl’s theorem, has been considered by a number of authors in the recent past
Summary
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 σb(T)of T coincides with the Weyl spectrum σw(T) of T; T satisfies Weyl’s theorem if the complement of σw(T) in σ (T) is the set (T) of finite multiplicity isolated eigenvalues of T. Let Xq := ∞(X )/m(X ), and denote by Sq the operator S∞ on Xq. The mapping S → Sq is a unital homomorphism from B(X ) → B(Xq) with kernel K(X ) which induces a norm decreasing monomorphism from B(X )/K(X ) to B(Xq) with the following properties (see [ , Section ] for details): (i) S is upper semi-Fredholm, S ∈ +, if and only if Sq is injective, if and only if Sq is bounded below; (ii) S is lower semi-Fredholm, S ∈ –, if and only if Sq is surjective; (iii) S is Fredholm, S ∈ , if and only if Sq is invertible.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
