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

Abstract

Given Banach space operators A ∈ B( Open image in new window ) and B ∈ B( Open image in new window ), let A⊗B ∈ B( Open image in new window ⊗ Open image in new window ) denote the tensor product of A and B. Let σa, σaw and σab denote the approximate point spectrum, the Weyl approximate point spectrum and the Browder approximate point spectrum, respectively. Then σaw(A⊗B) ⊆ σa(A)σaw(B) ⊂ σaw(A)σa(B) ⊆ σa(A)σab(B) ⊂ σab(A)σa(B) = σab(A⊗B), and a sufficient condition for the (a-Weyl spectrum) identity σaw(A⊗B) = σa(A)σaw(B) ⊂ σaw(A)σa(B) to hold is that σaw(A⊗B) = σab(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.