Acta Mathematica Vietnamica

, Volume 44, Issue 4, pp 993–1004 | Cite as

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

  • Mohammad H. M. RashidEmail author


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.


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

Mathematics Subject Classification (2010)

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



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


  1. 1.
    Aiena, P.: Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer Academic Publishers, Dordrecht (2004)zbMATHGoogle Scholar
  2. 2.
    Aiena, P., Carpintero, C., Rosas, E.: Some characterizations of operators satisfying a-Browder’s theorem. J. Math. Anal. Appl. 311(2), 530–544 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aiena, P., Peña, P.: Variations on Weyl’s theorem. J. Math. Anal. Appl. 324 (1), 566–579 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aiena, P., Triolo, S.: Fredholm spectra and Weyl type theorems for Drazin invertible operators. Mediterr. J. Math. 13(6), 4385–4400 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aiena, P., Triolo, S.: Local spectral theory for Drazin invertible operators. J. Math. Anal. Appl. 435(1), 414–424 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Aiena, P., Triolo, S.: Some perturbation results through localized SVEP. Acta Sci. Math. (Szeged) 82(1–2), 205–219 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Coburn, L.A.: Weyl’s theorem for nonnormal operators. Michigan Math. J. 13, 285–288 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Djordjević, D. S.: Perturbation of spectra of operator matrices. J. Operator Theory 48(3), 467–486 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Djordjević, S. V., Zguitti, H.: Essential point spectra of operator matrices through local spectral theory. J. Math. Anal. Appl. 338(1), 285–291 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duggal, B.P.: Hereditarily normaloid operators. Extracta Math. 20(2), 203–217 (2005)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Duggal, B.P.: Upper triangular operator matrices, SVEP and Browder, Weyl theorems. Integral Equ. Operator Theory 63(1), 17–28 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Duggal, B.P.: Browder and Weyl spectra of upper triangular operator matrices. Filomat 24(2), 111–130 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Heuser, H.: Functional Analysis. Wiley, Chichester (1982)zbMATHGoogle Scholar
  14. 14.
    Rakoc̆ević, V.: Operators obeying a-Weyl’s theorem. Rev. Roumaine Math. Pures Appl. 34(10), 915–919 (1989)MathSciNetGoogle Scholar
  15. 15.
    Rakoc̆ević, V.: On a class of operators. Mat. Vesnik 37(4), 423–426 (1985)MathSciNetGoogle Scholar
  16. 16.
    Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis. Wiley, New York (1980)zbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mathematics & Statistics, Faculty of Science P. O. Box (7)Mu’tah UniversityAl-KarakJordan

Personalised recommendations