Advances in Operator Theory

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

Anuradha Gupta and Karuna Mamtani

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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

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

Received: 3 January 2017
Accepted: 7 June 2017
First available in Project Euclid: 4 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]
Secondary: 47A10: Spectrum, resolvent 47A11: Local spectral properties

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


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.

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