# Variants of Weyl's theorem for direct sums of closed linear operators

Document Type: Original Article

Authors

University of Delhi, Delhi.

Abstract

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.

Keywords

### History

• Receive Date: 03 January 2017
• Revise Date: 05 June 2017
• Accept Date: 07 June 2017