Property (z); direct sums and a note on an a-Browder type theorem

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 \cite[Theorem 2.3]{SZ} 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.