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

## TL;DR

This paper characterizes properties (z) and (az) for operators with specific spectral conditions, examines their preservation under direct sums, and corrects a previous theorem with counterexamples.

## Contribution

It provides new characterizations of properties (z) and (az), analyzes their stability under direct sums, and corrects a prior incorrect proof regarding spectral equalities.

## Key findings

- Characterizations of properties (z) and (az) for operators with SVEP.
- Conditions for preservation of properties (z) and (az) under direct sums.
- Counterexamples showing the failure of a previously assumed spectral equality.

## 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.