  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 $B^*$.
