Tensor product of left polaroid operators
  A Banach space operator $T\in B(X)$ is left polaroid if for each $\lambda\in\hbox{iso}\sigma_a(T)$ there is an integer $d(\lambda)$ such that asc $(T-\lambda)=d(\lambda)<\infty$ and $(T-\lambda)^{d(\lambda)+1}X$ is closed; $T$ is finitely left polaroid if asc $(T-\lambda)<\infty$, $(T-\lambda)X$ is closed and $\dim(T-\lambda)^{-1}(0)<\infty$ at each $\lambda\in\hbox{iso }\sigma_a(T)$. 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 $\tau_{AB}$, 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 $0\not\in\hbox{iso}\sigma_a(A\otimes B)$; a similar result holds for $\tau_{AB}$ for finitely left polaroid $A$ and $B^*$. 
Enrico Boasso, B. P. Duggal
math.FA