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On the a-Browder and a-Weyl spectra of tensor products
Authors:Bhagwati P Duggal  Slavisa V Djordjević  Carlos S Kubrusly
Institution:1. 8 Redwood Grove, Northfield Avenue, London, W5 4SZ, UK
2. Benemerita Universidad Autonoma de Puebla, Puebla, Pue., 72570, Mexico
3. Catholic University of Rio de Janeiro, 22453-900, Rio de Janeiro, Brazil
Abstract:Given Banach space operators A ∈ B( ></img> </span> </span>) and <em>B</em> ∈ <em>B</em>(<span class= ></img> </span> </span>), let <em>A</em>?<em>B</em> ∈ <em>B</em>(<span class= ></img> </span> </span>?<span class= ></img> </span> </span>) denote the tensor product of <em>A</em> and <em>B</em>. Let σ<sub> <em>a</em> </sub>, σ<sub> <em>aw</em> </sub> and σ<sub> <em>ab</em> </sub> denote the approximate point spectrum, the Weyl approximate point spectrum and the Browder approximate point spectrum, respectively. Then σ<sub> <em>aw</em> </sub>(<em>A</em>?<em>B</em>) ? σ<sub> <em>a</em> </sub>(<em>A</em>)σ<sub> <em>aw</em> </sub>(<em>B</em>) ? σ<sub> <em>aw</em> </sub>(<em>A</em>)σ<sub> <em>a</em> </sub>(<em>B</em>) ? σ<sub> <em>a</em> </sub>(<em>A</em>)σ<sub> <em>ab</em> </sub>(<em>B</em>) ? σ<sub> <em>ab</em> </sub>(<em>A</em>)σ<sub> <em>a</em> </sub>(<em>B</em>) = σ<sub> <em>ab</em> </sub>(<em>A</em>?<em>B</em>), and a sufficient condition for the (<em>a</em>-Weyl spectrum) identity σ<sub> <em>aw</em> </sub>(<em>A</em>?<em>B</em>) = σ<sub> <em>a</em> </sub>(<em>A</em>)σ<sub> <em>aw</em> </sub>(<em>B</em>) ? σ<sub> <em>aw</em> </sub>(<em>A</em>)σ<sub> <em>a</em> </sub>(<em>B</em>) to hold is that σ<sub> <em>aw</em> </sub>(<em>A</em>?<em>B</em>) = σ<sub> <em>ab</em> </sub>(<em>A</em>?<em>B</em>). Equivalent conditions are proved in Theorem 1, and the problem of the transference of <em>a</em>-Weyl’s theorem for <em>a</em>-isoloid operators <em>A</em> and <em>B</em> to their tensor product <em>A</em>?<em>B</em> is considered in Theorem 2. Necessary and sufficient conditions for the (plain) Weyl spectrum identity are revisited in Theorem 3.</td> </tr> <tr> <td align=
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