Journal of Operator Theory

Volume 47, Issue 1, Winter 2002 pp. 79-95.

Tensor products and the semi-Browder joint spectra

Authors: Enrico Boasso
Author institution: Departamento de Matematica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pab. I, (1428) Buenos Aires, Argentina

Summary: 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,

$S=(S_1,\ldots,S_n)$ 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

$X_1\tilde{\otimes}X_2$ ,

$(S\otimes I,I\otimes T)= (S_1\otimes I,\ldots ,S_n\otimes I,I\otimes T_1,\ldots ,I \otimes T_m)$ , and we give a description of the semi-Browder joint spectra introduced by V.~Kordula, V.~M\"uller and V. Rako\v cevi\' c in [7] and of the split semi-Browder joint spectra (see Section 3) of the

$(n+m)$ -tuple

$(S\otimes I ,I\otimes T)$ , 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

$(S\otimes I ,I\otimes T)$ . 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].

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