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

ZARIOUH Hassan

Abstract


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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