Boundary values of semigroups of linear operators
by
Myroslav L. Gorbachuk
Institute of Mathematics, Ukrainian National Academy of Sciences

Let U(t) be a strongly continuous semigroup of bounded linear operators on a Banach space X. We denote by A its generator. It is obvious that for any x from X, the vector function U(t)x is a weak solution of the equation

y'(t) = Ay(t)

on the open half-axis t > 0. However, generally, not all weak solutions of this equation on the half-axis t > 0 can be written in the form y(t) = U(t)x with x from X. We indicate the space Y containing X, and the extension V(t) of the semigroup U(t) such that any weak solution of the above equation on the positive half-axis is represented by the formula y(t) = V(t)y where y belongs to Y. It is also developed the theory of boundary values of such solutions at the point 0, which contains, as a particular case, the corresponding theory for analytic functions (Fatou, Riesz, Köthe, Komatsu and some other theorems).

(P)

Date received: November 26, 1999