Property $(Z_{\pi_0^a})$ for direct sums



We show that generally the properties $(Z_{\pi_0^a})$ and $(Z_{p_0^a})$ introduced by the author are not
preserved under direct sum of operators. Moreover, If $S$ and $T$ are Banach spaces operators
satisfying property $(Z_{\pi_0^a})$ or $(Z_{p_0^a}),$ we give conditions on S and T to ensure the preservation
of these properties by the direct sum $S\oplus $ T.$ Some crucial applications are also given.

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P. Aiena, Fredholm and Local Spectral Theory, with Application to Multipliers, Kluwer Aca-

demic Publishers, (2004).


B. A. Barnes, Riesz points and Weyls theorem, Integral Equations Oper. Theory, 34 (1999),



A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integr. Equ. and Oper. Theory, 13

(1990), 307-315.


M. Berkani, M. Kachad, H. Zariouh, Extended Weyl-type theorems for direct sums, Demons.

Math., Vol. 47 (2) (2014), 411-422.


M. Berkani, M. Sarih, On semi B-Fredholm operators, Glasgow Math. J. 43 (2001), 457-465.


M. Berkani and H. Zariouh, New extended Weyl type theorems, Mat. Vesnik Vol. 62 (2) (2010),



M. Berkani, H. Zariouh, Weyl-type theorems for direct sums, Bull. Korean. Math. Soc. 49

(2012), No. 5, pp. 1027-1040.

J. B. Conway, (1990). The theory of subnormal operators, Mathematical Surveys and mlono-

graphs, N. 36, (1992). American Mathematical Society, Providence, Rhode Island. Springer-

Verlag, New York.


S. V. Djordjevic and Y. M. Han, A note on Weyl's theorem for operator matrices, Proc. Amer.

Math. Soc. 131, No. 8 (2003), pp. 2543-2547.


B. P. Duggal, C. S. Kubrusly, Weyl's theorem for direct sums, Studia Sci. Math. Hungar. 44

(2007), 275-290.


H. Heuser, Functional Analysis, John Wiley & Sons Inc, New York, (1982).


W. Y. Lee, Weyl spectra of operator matrices, Proc. Amer. Math. Soc. 129 (2001), 131-138.


K. B. Laursen and M. M. Neumann, An introduction to Local Spectral Theory, Clarendon

Press Oxford, (2000).


H. Zariouh, On the property (ZEa ), Rend. Circ. Mat. Palermo, Rend. Circ. Mat. Palermo 65

(2016), 323-331.


H. Zariouh and Zguitti, Variations on Browder's theorem, Acta Math. Univ. Comenianae Vol

, 2 (2012), pp, 255-264.


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