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

Abstract

When $A\in \mathscr{L}(\mathbb {X})$ and $B\in \mathscr{L}(\mathbb {Y})$ are given, we denote by MC an operator acting on the Banach space $\mathbb {X}\oplus \mathbb {Y}$ of the form $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 $\mathbf {\sigma }_{aw}(M_{C})\subseteq \mathbf {\sigma }_{aw}(M_{0})\cup S_{+}^{*}(A)\cup S_{+}(B)$ . 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 $E^{0}(M_{C})=\{\lambda \in \text {iso}\sigma (M_{C}):0<\dim (M_{C}-\lambda )^{-1}\}$ , M0 satisfies property (w), and A and B satisfy either the hypotheses (i) A has SVEP at points $\mathbf {\lambda }\in \mathbf {\sigma }_{aw}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)$ and A∗ has SVEP at points $\mu \in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)$ , or (ii) A∗ has SVEP at points $\mathbf {\lambda }\in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)$ and B∗ has SVEP at points $\mu \in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(B)$ , 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 $\mathbf {\lambda }\in {E_{a}^{0}}(M_{C})$ are poles of A and points $\mu \in {E_{a}^{0}}(B)$ are poles of B, then MC satisfies property (w). Also, we give an example to illustrate our results.