Abstract. A Banach space operator $T\in B({\cal X})$ is left polaroid if for each $\lambda\in \mathop{\rm iso} \sigma_a(T)$ there is an integer $d(\lambda )$ such that $\mathop{\rm asc} (T-\lambda )=d(\lambda )< \infty $ and $(T-\lambda )^{d(\lambda )+1}{\cal X}$ is closed; $T$ is finitely left polaroid if $\mathop{\rm asc} (T-\lambda )< \infty $, $(T-\lambda ){\cal X}$ is closed and $\dim(T-\lambda )^{-1}(0)< \infty $ at each $\lambda\in \mathop{\rm iso} \sigma_a(T)$. 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 $0\not\in\mathop{\rm iso} \sigma_a(A\otimes B)$; a similar result holds for $\tau_{AB}$ for finitely left polaroid $A$ and $B^*$.

AMS Subject Classification (1991): 47A80, 47A53, 47A10

Keyword(s): Banach space, left polaroid operator, finitely left polaroid operator, tensor product, left-right multiplication, generalized $a$-Weyl's theorem

Received December 6, 2010. (Registered under 84/2010.)