The joint spectrum of commuting operators, as introduced by Taylor, has been shown by Cho and Takaguchi to be the set of joint eigenvalues in the case of matrices. These joint eigenvalues can be read off from their simultaneous uppertriangularization. We prove here a similar result for compact operators on Banach spaces.
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Tirthankar Bhattacharyya, On Tuples of Commuting Compact Operators. Publ. Res. Inst. Math. Sci. 32 (1996), no. 5, pp. 785–795DOI 10.2977/PRIMS/1195162382