TENSOR PRODUCTS AND THE SEMI-BROWDER JOINT SPECTRA

Abstract. Given two complex Banach spaces X1 and X2, a tensor product of X1 and X2, X1e⊗X2, in the sense of J. Eschmeier ([5]), and two finite tuples of commuting operators, S = (S1,..., Sn) and T = (T1,..., Tm), defined on X1 and X2 respectively, we consider the (n+m)-tuple of operators defined on X1e⊗X2, (S ⊗ I, I⊗T) = (S1 ⊗ I,..., Sn ⊗ I, I⊗T1,..., I⊗Tm), and we give a description of the semi-Browder joint spectra introduced by V. Kordula, V. Müller and V. Rakočevic ́ in [7] and of the split semi-Browder joint spectra (see Section 3) of the (n+m)-tuple (S⊗I, I⊗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 (S ⊗ I, I ⊗ 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]