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

Research paper by Mohammad H. M. Rashid

Indexed on: 09 Mar '19Published on: 06 Mar '19Published in: Acta Mathematica Vietnamica

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.