Browder–Weyl theorems, tensor products and multiplications

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A Banach space operator T ∈ B ( X ) satisfies Browder's theorem if the complement of the Weyl spectrum σ w ( T ) of T in σ ( T ) equals the set of Riesz points of T ; T is polaroid if the isolated points of σ ( T ) are poles (no restriction on rank) of the resolvent of T . Let Φ ( T ) denote the set of Fredholm points of T . Browder's theorem transfers from A , B ∈ B ( X ) to S = L A R B (resp., S = A ⊗ B ) if and only if A and B ∗ (resp., A and B ) have SVEP at points μ ∈ Φ ( A ) and ν ∈ Φ ( B ) for which λ = μ ν ∉ σ w ( S ) . If A and B are finitely polaroid, then the polaroid property transfers from A ∈ B ( X ) and B ∈ B ( Y ) to L A R B ; again, restricting ourselves to the completion of X ⊗ Y in the projective topology, if A and B are finitely polaroid, then the polaroid property transfers from A ∈ B ( X ) and B ∈ B ( Y ) to A ⊗ B .