Tensor product of left polaroid operators

A Banach space operator T ∈ B(χ) is left polaroid if for each λ ∈ iso σa(T) there is an integer d(λ) such that asc(T - λ) = d(λ) < ∞ and (T - λ)d(λ)+1χ is closed; T is finitely left polaroid if asc(T - λ) < ∞, (T - λ)χ is closed and dim(T - λ)−1(0) < ∞ at each λ ∈ iso 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 0 ∈ iso σa(A⊗B); a similar result holds for τAB for finitely left polaroid A and B*.