Browder spectra for upper triangular operator matrices

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the infinite dimensional separable Hilbert space H⊕K of the form . In this paper, we prove that

     

Browder spectra and essential spectra of operator matrices

Let Mc = （ A0CB ） be a 2 × 2 upper triangular operator matrix acting on the Banach space X × Y. We prove that
σr（A） ∪ σr（ B）= σr （Mc） ∪ W ,
where W is the union of certain of the holes in σr（Mc） which happen to be subsets of σr（A） ∩ σr（B）, and σr（A）, σr（B）, σr（Mc） can be equal to the Browder or essential spectra of A, B, Mc, respectively. We also show that the above result isn＇t true for the Kato spectrum, left （right） essential spectrum and left （right） spectrum.     

Browder spectra of upper-triangular operator matrices

Let be a 2×2 upper triangular operator matrix acting on the Hilbert space H⊕K. In this paper, for given operators A and B, we prove that

     

Generalized Drazin spectrum of operator matrices

Let A ∈ B(X) and B ∈ B(Y), MC be an operator on Banach space X ⊕ Y given A C by MC =A generalized Drazin spectrum defined by σgD(T) = {λ∈ C : T-0 BλI is not generalized Drazin invertible} is considered in this paperIt is shown thatσgD(A) ∪σgD(B) = σgD(MC) ∪ WgD(A, B, C),where WgD(A, B, C) is a subset of σgD(A) ∩σgD(B) and a union of certain holes in σgD(MC).Furthermore, several sufficient conditions for σgD(A) ∪σgD(B) = σgD(MC) holds for every C ∈ B(Y, X) are given.     

Continuity of spectrum and approximate point spectrum on operator matrices

Let X and Y be given Banach spaces. For A∈B(X), B∈B(Y) and C∈B(Y,X), let MC be the operator defined on X⊕Y by . In this paper we give conditions for continuity of τ at MC through continuity of τ at A and B, where τ can be equal to the spectrum or approximate point spectrum.     

Continuous spectrum, point spectrum and residual spectrum of operator matrices

Let M_C denote a 2 × 2 upper triangular operator matrix of the form , which is acting on the sum of Banach spaces X⊕Y or Hilbert spaces H⊕K. In this paper, the sets and ?_C∈B(K,H)σ_r(M_C) are, respectively, characterized completely, where σ_c(·) denotes the continuous spectrum, σ_p(·) denotes the point spectrum and σ_r(·) denotes the residual spectrum. Moreover, some corresponding counterexamples are given.     

算子的Kato本质谱与Browder本质谱的连通分支

利用算子谱的精密结构的分析方法,给出算子S和T拟相似时Kato本质谱σK(S)的一个连通分支与σK(T)相交的充分条件和充要条件,同时证明了Browder本质谱σB(S)的每一个连通分支与Kato本质谱σK(T)的某些子集的交集是非空的.     

Essential,Weyl and Browder spectra of unbounded upper triangular operator matrices

This paper is concerned with the spectral properties of the unbounded upper triangular operator matrix with diagonal domain. Some sufficient and necessary conditions are given under which the essential spectrum, the Weyl spectrum and the Browder spectrum of such operator matrix, respectively, coincide with the union of the essential spectrum, the Weyl spectrum and the Browder spectrum of its diagonal entries.     

Idempotent completions of operator partial matrices

Necessary and sufficient conditions are obtained for operator partial 2 × 2 matrices to have an idempotent completion, and all such completions are parametrically represented. Project supported by National Natural Science Foundation of China and Provincial Natural Science Foundation of Shanxi     

On the Drazin inverse of the sum of two operators and its application to operator matrices

Given two bounded linear operators F,G on a Banach space X such that G²F=GF²=0, we derive an explicit expression for the Drazin inverse of F+G. For this purpose, firstly, we obtain a formula for the resolvent of an auxiliary operator matrix in the form . From the provided representation of D(F+G) several special cases are considered. In particular, we recover the case GF=0 studied by Hartwig et al. [R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207-217] for matrices and by Djordjevi? and Wei [D.S. Djordjevi?, Y. Wei, Additive results for the generalized Drazin inverse, J. Aust. Math. Soc. 73 (1) (2002) 115-126] for operators. Finally, we apply our results to obtain representations for the Drazin inverse of operator matrices in the form which are extensions of some cases given in the literature.     

A note on the representation for the Drazin inverse of block matrices

In this note we consider representations of the Drazin inverse of 2×2 block matrices under conditions weaker than those used in recent papers on the subject, in particular in [D.S. Djordjević, P.S. Stanimirović, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (126) (2001) 617–634; R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of 2×2 block matrix, SIAM J. Matrix Anal. Appl. 27 (2006) 757–771].     

Drazin invertibility of upper triangular operator matrices

In this paper, the existence is considered of a Drazin invertible completion of an upper triangular operator matrix. The proof of a recently published result is corrected.     

On Drazin inverse of operator matrices

In this short paper, we offer (another) formula for the Drazin inverse of an operator matrix for which certain products of the entries vanish. We also give formula for the Drazin inverse of the sum of two operators under special conditions.     

A note on Faber operator

For a rectifable Jordan curve Γ with complementary domainsD and D,Anderson conjectured that the Faber operator is a bounded isomorphism between the Besov spaces Bp(1 p ∞) of analytic functions in the unit disk and in the inner domain D,respectively.We point out that the conjecture is not true in the special case p=2,and show that in this case the Faber operator is a bounded isomorphism if and only if Γ is a quasi-circle.