(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 1, PAGES 1-16

## Splitting of perturbated differential operators with unbounded operator coefficients

Abstract

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We obtain some theorems on splitting of differential operators of the form

$\mathcal L = \frac\left\{d\right\}\left\{dt\right\} - A_0 - B A_0^\nu \colon$
D(\mathcal L) \subset C(\mathbb R,\mathcal Y) \to C(\mathbb R,\mathcal Y) 

acting in the Banach space $C\left(\mathbb R,\mathcal Y\right)$ of continuous and bounded functions defined on real axis R with values in the Banach space $\mathcal Y$. The linear operator $A_0 \colon D\left(A_0\right) \subset \mathcal Y \to \mathcal Y$ is the generating operator of a strongly continuous semigroup of operators and its spectrum does not intersect the imaginary axis $i$R. Here $A$0n, n Î [0,1), is a fractional power of $A$0 and $B \colon C\left(\mathbb R,\mathcal Y\right) \to C\left(\mathbb R,\mathcal Y\right)$ is a bounded linear operator.

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