Tensor product of left polaroid operators

Abstract

A Banach space operator $T\in B(X)$ is left polaroid if for each $\lambda \in iso{\sigma }_a(T)$ 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 $\lambda \in iso{\sigma }_a(T)$. The left polaroid property transfers from $A$ and $B$ to their tensor product $A\otimes B$, hence also from $A$ and $B^{\ast }$ 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 $0\notin iso{\sigma }_a(A\otimes B)$; a similar result holds for ${\tau }_{AB}$ for finitely left polaroid $A$ and $B^{\ast }$.

arXiv comment10 pages, original research article