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Given unital Banach algebras $A$ and $B$ and elements $a\in A$ and $b\in B$,the Drazin spectrun of $a\otimes b\in A\overline{\otimes} B$ will be fullycharacterized, where $A\overline{\otimes} B$ is a Banach algebra that is thecompletion of $A\otimes B$ with respect to a uniform crossnorm. To this end,however, first the isolated points of the spectrum of $a\otimes b\inA\overline{\otimes} B$ need to be characterized. On the other hand, givenBanach spaces $X$ and $Y$ and Banach space operators $S\in L(X)$ and $T\inL(Y)$, using similar arguments the Drazin spectrum of $\tau_{ST}\in L(L(Y,X))$,the elementary operator defined by $S$ and $T$, will be fully characterized.

The Drazin spectrum of tensor product of Banach algebra elements and elementary operators