  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.
