If $T$ is an operator with compact resolvent and $S$ is any densely defined closed linear operator, then the orthogonal direct sum of $T$ and $S$ satisfies various Weyl type theorems if some necessary conditions are imposed on the operator $S$. It is shown that if $S$ is isoloid and satisfies Weyl's theorem, then $T \oplus S$ satisfies Weyl's theorem. Analogous result is proved for a-Weyl's theorem. Further, it is shown that Browder's theorem is directly transmitted from $S$ to $T \oplus S$. The converse of these results have also been studied.
"Variants of Weyl's theorem for direct sums of closed linear operators." Adv. Oper. Theory 2 (4) 409 - 418, Autumn 2017. https://doi.org/10.22034/aot.1701-1087