## Advances in Operator Theory

- Adv. Oper. Theory
- Volume 2, Number 4 (2017), 409-418.

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

Anuradha Gupta and Karuna Mamtani

#### 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.

#### Article information

**Source**

Adv. Oper. Theory, Volume 2, Number 4 (2017), 409-418.

**Dates**

Received: 3 January 2017

Accepted: 7 June 2017

First available in Project Euclid: 4 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aot/1512431717

**Digital Object Identifier**

doi:10.22034/aot.1701-1087

**Mathematical Reviews number (MathSciNet)**

MR3730036

**Zentralblatt MATH identifier**

06804217

**Subjects**

Primary: 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]

Secondary: 47A10: Spectrum, resolvent 47A11: Local spectral properties

**Keywords**

operators with compact resolvent direct sums Weyl’s Theorem a-Weyl’s Theorem Browder’s Theorem

#### Citation

Gupta, Anuradha; Mamtani, Karuna. Variants of Weyl's theorem for direct sums of closed linear operators. Adv. Oper. Theory 2 (2017), no. 4, 409--418. doi:10.22034/aot.1701-1087. https://projecteuclid.org/euclid.aot/1512431717