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We give a simple proof of a general theorem of Kotake-Narasimhan for elliptic operators in the setting of ultradifferentiable functions in the sense of Braun, Meise and Taylor. We follow the ideas of Komatsu. Based on an example of Métivier, we also show that the ellipticity is a necessary condition for the theorem to be true. [ABSTRACT FROM AUTHOR]

We study $$p$$-evolution pseudo-differential systems of the first order with coefficients in $$(t,x)$$ and real characteristics. We find sufficient conditions for the well-posedness of the Cauchy problem in $$H^\infty $$. These conditions involve the behavior as $$x\rightarrow \infty $$ of the coefficients, requiring some decay estimates to be satisfied. [ABSTRACT FROM AUTHOR]