Tensor product of left polaroid operators

A Banach space operator T is an element of B(X) is left polaroid if for each lambda is an element of iso sigma(a)(T) there is an integer d(lambda) such that asc(T - lambda) = d(lambda) < infinity and (T - lambda) d((lambda)+ 1) X is closed; T is finitely left polaroid if asc(T - lambda) < infinity, (T - lambda) infinity is closed and dim(T - lambda)(-1)(0) < infinity at each lambda is an element of iso sigma(a)(T). The left polaroid property transfers from A and B to their tensor product A circle times B, hence also from A and B* to the left-right multiplication operator T-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 circle times B if and only if 0 is not an element of iso sigma(a)(A circle times B); a similar result holds for T-AB for finitely left polaroid A and B*.