Tensor products and the semi-Browder joint spectra

Given two complex Banach spaces $X_1$ and $X_2$, a tensor product of $X_1$ and $X_2$, $X_1\tilde{\otimes }{X}_2$, in the sense of J. Eschmeier ([5]), and two finite tuples of commuting operators, $S=(S_1,\dots ,S_n)$ and $T=(T_1,\ldots ,T_m)$,definedon$X_1$and$X_2$respectively,weconsiderthe$(n+m)$-tuple of operators defined on $X_1\tilde{\otimes }{X}_2$, $(S\otimes I,I\otimes T)= (S_1\otimes I,\ldots ,S_n\otimes I,I\otimes T_1,\ldots ,I \otimes T_m)$, and we give a description of the semi-Browder joint spectra introduced by V. Kordula, V. M"uller and V. Rako$\check{c}$evi$\acute{c}$ in [7] and of the split semi-Browder joint spectra (see section 3), of the $(n+m)$-tuple $(S\otimes I ,I\otimes T)$,intermsofthecorrespondingjointspectraof$S$and$T$. This result is in some sense a generalization of a formula obtained for other various Browder spectra in Hilbert spaces and for tensor products of operators and for tuples of the form $(S\otimes I,I\otimes T)$. In addition, we also describe all the mentioned joint spectra for a tuple of left and right multiplications defined on an operator ideal between Banach spaces in the sense of [5].