Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $B^*$.

4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc