Linear Operator Pencils
In this chapter we study linear pencils λG − A, where G and A are bounded linear operators acting between complex Banach spaces X and Y. The simplest example is the case when X = Y and G = I, the identity operator on X. This case was already considered in the previous chapters. If G is invertible, then spectral problems concerning the pencil λG − A are easily reduced to those of λI − G −1 A. In this chapter we consider the more general case when G is not invertible, but we assume that the pencil is regular, i.e., for some λ 0 ∈ ℂ the operator λ 0 G − A is invertible.
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