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 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.

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Acknowledgements

The author take this opportunity to thank the referee for his very helpful comments on the submitted version of the paper.

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Correspondence to Mohammad H. M. Rashid.

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Rashid, M.H.M. Upper Triangular Operator Matrices, SVEP, and Property (w). Acta Math Vietnam 44, 993–1004 (2019). https://doi.org/10.1007/s40306-018-00307-0

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Keywords

  • Weyl’s theorem
  • Weyl spectrum
  • Polaroid operators
  • Property (w)
  • Upper triangular operator matrices
  • SVEP

Mathematics Subject Classification (2010)

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