International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 9, Pages 555-563

On an abstract evolution equation with a spectral operator of scalar type

Marat V. Markin

Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston 02215, MA, USA

Received 20 December 2001

Copyright © 2002 Marat V. Markin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


It is shown that the weak solutions of the evolution equation y(t)=Ay(t), t[0,T)(0<T), where A is a spectral operator of scalar type in a complex Banach space X, defined by Ball (1977), are given by the formula y(t)=etAf, t[0,T), with the exponentials understood in the sense of the operational calculus for such operators and the set of the initial values, f's, being 0t<TD(etA), that is, the largest possible such a set in X.