Groups of weighted translation operators in \(L_ p\) and linear nonautonomous differential equations (Q1900379)

This paper is concerned with linear nonautonomous differential equations of the form \(x'= A(t)x\), defined on the real line. The main result gives necessary and sufficient conditions for the underlying equation to have an exponential dichotomy, in terms of spectral properties of the differential operator \(- d/dt+ A(t)\) and the one-parameter semigroup of linear operators generated by it as operators acting in the function spaces \(L_p\) \((1\leq p< \infty)\). The obtained results are a generalization of previous work by the second author [see e.g. Trans. Am. Math. Soc. 345, No. 1, 223-241 (1994; Zbl 0820.34039), and J. Math. Anal. Appl. 187, No. 2, 339-351 (1994; Zbl 0811.34064)].