Some operators acting on weighted sequence spaces and applications

This paper considers the problem of constructing an evolution family for the linear nonautonomous Cauchy problem: () @ @t u(t) ?? A(t)u(t) = 0; u(??1) = x 2 RN; where A 2 C([??1; 1], RNN). The essence of the method is that the evolution family is sought in the form of a series of Chebyshev polynomials. Then, by dening appropriate weighted sequence spaces and matrices of linear operators, we are able to obtain a sucient condition - based only in the given data - for the representation of the solution of the initial value problem (*). The method is motivated for practical considerations in the context of Magnetic Resonance Imaging.