IJPAM: Volume 118, No. 3 (2018)

Title

PROPERTY $(z),$ DIRECT SUMS AND
A NOTE ON AN A-BROWDER TYPE THEOREM

Authors

Abdelmajid Arroud$^1$, Hassan Zariouh$^2$
Laboratory (L.A.N.O)
Department of Mathematics
Faculty of Science, Mohammed I University
PO Box 717, Oujda 60000, MOROCCO
Department of Mathematics (CRMEFO)
and Laboratory (L.A.N.O)
Faculty of Science
Mohammed I University
Oujda, 60000, MOROCCO

Abstract

We characterize the properties $(z)$ and $(az)$ for an operator $T$ whose dual $T^*$ has the SVEP on the complementary of the upper semi-Weyl spectrum of $T.$ If $S$ and $T$ are Banach space operators satisfying property $(z)$ or $(az),$ we give conditions on $S$ and $T$ to ensure the preservation of these properties by the direct sum $S\oplus T.$ Some results are given for multipliers and in general for $(H)$-operators. Also we give a correct proof of [11, Theorem 2.3] which was proved by using the equality $\sigma_p^0(S\oplus T)= \sigma_p^0(S)\cup \sigma_p^0(T).$ However this equality is not true; we give counterexamples to show that.

History

Received: August 1, 2017
Revised: January 19, 2018
Published: April 11, 2018

AMS Classification, Key Words

AMS Subject Classification: 47A53, 47A10, 47A11
Key Words and Phrases: property $(z)$, property $(az)$, upper semi-Weyl spectrum, direct sum

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Bibliography

1
P. AIENA, Fredholm and Local Spectral Theory, with Application to Multipliers, Kluwer Academic Publishers, Dordrecht (2004). isbn: 1-4020-1830-4

2
P. AIENA, Property (w) and perturbations II, J. Math. Anal. Appl., 342(2), (2008), 830-837. https://doi.org/10.1016/j.jmaa.2007.12.029

3
P. AIENA, P. PEA, Variations on Weyl's theorem, J. Math. Anal. Appl., 324(1), (2006), 566-579. https://doi.org/10.1016/j.jmaa.2005.11.027

4
P. AIENA, J.R. GUILLEN, Weyl's theorem for perturbations of paranormal operators, Proc. Amer. Math. Soc., 135(8), (2007), 2443-2451. https://doi.org/10.1090/S0002-9939-07-08582-6

5
P. AIENA, J.R. GUILLEN, P. PEA, Property (w) for perturbations of polaroid operators, Linear. Alg. App., 428(8-9), (2008), 1791-1802. https://doi.org/10.1016/j.laa.2007.10.022

6
P. AIENA, K.B. LAURSEN, Multipliers with closed range on regular commutative Banach algebras, Proc. Am. Math. Soc., 121(4), (1994), 1039-1048. https://doi.org/10.1090/S0002-9939-1994-1185257-1

7
A. ARROUD, H. ZARIOUH, Browder-type theorems for direct sums of operators, Func. Anal. Approx. Comp., 7(1), (2015), 77-84.

8
A. ARROUD, H. ZARIOUH, Stability of Browder spectral properties under Riesz-type perturbations, Func. Anal. Approx. Comp., 7(3), (2015), 57-65.

9
B.A. BARNES, Riesz points and Weyl's theorem, Integral Equations Oper. Theory, 34(2), (1999), 187-196. https://doi.org/10.1007/BF01236471

10
M.BERKANI, M.KACHAD, H. ZARIOUH, H. ZGUITTI, Variations on a-Browder-type theorems, Sarajevo J. Math., 9(2), (2013), 271-281. doi: 10.5644/SJM.09.2.11

11
M. BERKANI, M. SARIH, H. ZARIOUH, a-Browder-type theorems for direct sums of operators, Math. Bohemica, 141(1), (2016), 99-108. doi: 10.21136/MB.2016.8

12
R.E. CURTO, Y.M. HAN, Weyl's theorem, a-Weyl's theorem, and local spectral theory, J. London Math. Soc., 67(2), (2003), 499-509. https://doi.org/10.1112/S0024610702004027

13
R.E. CURTO, Y.M. HAN, Weyl's theorem for algebraically paranormal operators. In- tegr. equ. oper. theory, 47(3), (2003), 307-314. doi: 10.1007/s00020-002-1164-1

14
B.P. DUGGAL, C.S. KUBRUSLY, Weyl's theorem for direct sums, Studia Scientiarum Mathematicarum Hungarica, 44(2), (2007), 275-290. doi: 10.1556/SScMath.2007.1018

15
A.GUPTA, N.KASHYAP, Generalized a-Weyl’s theorem for direct sums, Matematicki Vesnik, 62(4), (2010), 265-270.

16
K.B. LAURSEN, M.M. NEUMANN, An introduction to Local Spectral Theory, Clarendon Press, Oxford (2000). isbn: 0198523815

17
W.Y. LEE, Weyl spectra of operator matrices, Proc. Amer. Math. Soc., 129(1), (2001), 131-138. https://doi.org/10.1090/S0002-9939-00-05846-9

18
V. RAKOEVI, Operators obeying a-Weyl's theorem, Rev. Roumaine Math. Pures Appl., 34(10), (1989), 915-919.

19
H. ZARIOUH, Property $(gz)$ for bounded linear operators, Mat. Vesnik, 65(1), (2013), 94-103.

How to Cite?

DOI: 10.12732/ijpam.v118i3.8 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 118
Issue: 3
Pages: 581 - 597


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