Tensor product of left polaroid operators

A Banach space operator $T\in B(X)$ is left polaroid if for each $λ\in\hbox{iso}σ_a(T)$ there is an integer $d(λ)$ such that asc $(T-λ)=d(λ)<\infty$ and $(T-λ)^{d(λ)+1}X$ is closed; $T$ is finitely left polaroid if asc $(T-λ)<\infty$, $(T-λ)X$ is closed and $\dim(T-λ)^{-1}(0)<\infty$ at each $λ\in\hbox{iso }σ_a(T)$. The left polaroid property transfers from $A$ and $B$ to their tensor product $A\otimes B$, hence also from $A$ and $B^*$ to the left-right multiplication operator $τ_{AB}$, 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 $A\otimes B$ if and only if $0\not\in\hbox{iso}σ_a(A\otimes B)$; a similar result holds for $τ_{AB}$ for finitely left polaroid $A$ and $B^*$.

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