Acta Mathematica Vietnamica

, Volume 44, Issue 4, pp 993–1004

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

Article

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 λE0(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.

Keywords

Weyl’s theorem Weyl spectrum Polaroid operators Property (wUpper triangular operator matrices SVEP

Mathematics Subject Classification (2010)

Primary 47A55 47A53 47B20 Secondary 47A10 47A11

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