Upper Triangular Operator Matrices, SVEP, and Property (w)

Abstract

When A∈L(X)$A\\in \\mathscr{L}(\\mathbb {X})$ and B∈L(Y)$B\\in \\mathscr{L}(\\mathbb {Y})$ are given, we denote by MC an operator acting on the Banach space X⊕Y$\\mathbb {X}\\oplus \\mathbb {Y}$ of the form MC=AC0B$M_{C}=\\left (\\begin {array}{cccccccc} A & C \\\\ 0 & B \\\\ \\end {array}\\right ) $. In this paper, first we prove that σw(M0) = σw(MC) ∪{S(A∗) ∩ S(B)} and σaw(MC)⊆σaw(M0)∪S+∗(A)∪S+(B)$\\mathbf {\\sigma }_{aw}(M_{C})\\subseteq \\mathbf {\\sigma }_{aw}(M_{0})\\cup S_{+}^{*}(A)\\cup S_{+}(B)$. Also, we give the necessary and sufficient condition for MC to be obeys property (w). Moreover, we explore how property (w) survive for 2 × 2 upper triangular operator matrices MC. In fact, we prove that if A is polaroid on E0(MC)={λ∈isoσ(MC):0<dim(MC−λ)−1}$E^{0}(M_{C})=\\{\\lambda \\in \\text {iso}\\sigma (M_{C}):0<\\dim (M_{C}-\\lambda )^{-1}\\}$, M0 satisfies property (w), and A and B satisfy either the hypotheses (i) A has SVEP at points λ∈σaw(M0)∖σSF+(A)$\\mathbf {\\lambda }\\in \\mathbf {\\sigma }_{aw}(M_{0})\\setminus \\mathbf {\\sigma }_{SF_{+}}(A)$ and A∗ has SVEP at points μ∈σw(M0)∖σSF+(A)$\\mu \\in \\mathbf {\\sigma }_{w}(M_{0})\\setminus \\mathbf {\\sigma }_{SF_{+}}(A)$, or (ii) A∗ has SVEP at points λ∈σw(M0)∖σSF+(A)$\\mathbf {\\lambda }\\in \\mathbf {\\sigma }_{w}(M_{0})\\setminus \\mathbf {\\sigma }_{SF_{+}}(A)$ and B∗ has SVEP at points μ∈σw(M0)∖σSF+(B)$\\mu \\in \\mathbf {\\sigma }_{w}(M_{0})\\setminus \\mathbf {\\sigma }_{SF_{+}}(B)$, then MC satisfies property (w). Here, the hypothesis that points λ ∈ E0(MC) are poles of A is essential. We prove also that if S(A∗) ∪ S(B∗), points λ∈Ea0(MC)$\\mathbf {\\lambda }\\in {E_{a}^{0}}(M_{C})$ are poles of A and points μ∈Ea0(B)$\\mu \\in {E_{a}^{0}}(B)$ are poles of B, then MC satisfies property (w). Also, we give an example to illustrate our results.