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On the a-Browder and a-Weyl spectra of tensor products

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Abstract

Given Banach space operators A ∈ B(

) and B ∈ B(

), let A⊗B ∈ B(

⊗

) denote the tensor product of A and B. Let σ a , σ aw and σ ab denote the approximate point spectrum, the Weyl approximate point spectrum and the Browder approximate point spectrum, respectively. Then σ aw (A⊗B) ⊆ σ a (A)σ aw (B) ⊂ σ aw (A)σ a (B) ⊆ σ a (A)σ ab (B) ⊂ σ ab (A)σ a (B) = σ ab (A⊗B), and a sufficient condition for the (a-Weyl spectrum) identity σ aw (A⊗B) = σ a (A)σ aw (B) ⊂ σ aw (A)σ a (B) to hold is that σ aw (A⊗B) = σ ab (A⊗B). Equivalent conditions are proved in Theorem 1, and the problem of the transference of a-Weyl’s theorem for a-isoloid operators A and B to their tensor product A⊗B is considered in Theorem 2. Necessary and sufficient conditions for the (plain) Weyl spectrum identity are revisited in Theorem 3.

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Authors and Affiliations

  1. 8 Redwood Grove, Northfield Avenue, London, W5 4SZ, UK

    Bhagwati P. Duggal

  2. Benemerita Universidad Autonoma de Puebla, Puebla, Pue., 72570, Mexico

    Slavisa V. Djordjević

  3. Catholic University of Rio de Janeiro, 22453-900, Rio de Janeiro, Brazil

    Carlos S. Kubrusly

Authors
  1. Bhagwati P. Duggal
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  2. Slavisa V. Djordjević
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  3. Carlos S. Kubrusly
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Correspondence to Carlos S. Kubrusly.

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Duggal, B.P., Djordjević, S.V. & Kubrusly, C.S. On the a-Browder and a-Weyl spectra of tensor products. Rend. Circ. Mat. Palermo 59, 473–481 (2010). https://doi.org/10.1007/s12215-010-0035-x

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  • DOI: https://doi.org/10.1007/s12215-010-0035-x

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