Tensor product of left polaroid operators

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A Banach space operator T∈ B(X) is left polaroid if for each λσₐ(T) there is an integer d(λ) such that asc (T-λ)=d(λ)<∞ and (T-λ)ᵈ⁽λ⁾⁺¹X is closed; T is finitely left polaroid if asc (T-λ)<∞, (T-λ)X is closed and (T-λ)⁻¹(0)<∞ at each λ σₐ(T). The left polaroid property transfers from A and B to their tensor product A⊗ B, hence also from A and B* to the left-right multiplication operator τAB, for Hilbert space operators; an additional condition is required for Banach space operators. The finitely left polaroid property transfers from A and B to their tensor product A⊗ B if and only if 0σₐ(A⊗ B); a similar result holds for τAB for finitely left polaroid A and B*.