Given two complex Banach spaces $X_1$ and $X_2$, a tensor product of $X_1$ and $X_2$, $X_1tilde{otimes}X_2$, in the sense of J. Eschmeier ([5]), and two finite tuples of commuting operators, $S=(S_1,ldots ,S_n)$ and $T=(T_1,ldots ,T_m)$, defined on $X_1$ and $X_2$ respectively, we consider the $(n+m)$-tuple of operators defined on $X_1tilde{otimes}X_2$, $(Sotimes I,Iotimes T)= (S_1otimes I,ldots ,S_notimes I,Iotimes T_1,ldots ,I otimes T_m)$, and we give a description of the semi-Browder joint spectra introduced by V. Kordula, V. Muller 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 $(Sotimes I ,Iotimes T)$, in terms of the corresponding joint spectra of $S$ 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 $(Sotimes I ,Iotimes T)$. 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].
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