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Ordered pairs of linear operators (A, B) from Banach space L (X; Y ) of the linear bounded operators, defined on complex Banach space X with the values in complex Banach space Y are considered in this paper. Notion of singular set (spectrum) of the ordered pair (A, B) is introduced essentially with the help of operators bundle of the form A − λB, λ ∈ C,

Spectral theory of ordered pairs of the linear operators - acting in different Banach spaces and applications