Consider the differential
equation
 (1.1) 
where u(t), f(t) are elements of a Hilbert space E and A(t) is a closed linear operator
in E with a domain D(A) independent of t and dense in E. Denote by C^{m}(a,b) the
set of functions v(t) with values in E which have m strongly continuous derivatives in
(a,b). Introducing the norm
 (1.2) 
where v(t) is the Enorm of v(t), we denote by H^{m}(a,b) the completion with
respect to the norm (1.2) of the subset of functions in C^{m}(a,b) whose norm is finite.
Set H^{m} = H^{m}(−∞,∞) and denote by H_{0}^{m} the subset of functions in H^{m} which
have compact support. The solutions u(t) of (1.1) are understood in the sense that
u(t) ∈ H^{1}(a′,b′) for any a < a′ < b′ < b.
Theorem 1. Assume that, for each a < t < b, the resolvent R(λ,A(t)) = (λ−A(t))^{−1}
of A(t) exists for all real λ, λ≧ N(t), and that
 (1.3) 
where N(t), C(t) are constants. Assume next that for each s ∈ (a,b), A^{−1}(s) exists
and
 (1.4) 
for a < t < b, where m is any integer ≧ 1. If u is a solution of (1.1) and if
f ∈ H^{m}(a,b), then u ∈ H^{m+1}(a′,b′) for any a < a′ < b′ < b.
Theorem 2. If the assumptions of Theorem 1 hold with m = ∞, if A(t)A^{−1}(s) is
analytic in t(a < t < b) for each s ∈ (a,b), and if f(t) is analytic in (a,b), then u(t) is
also analytic in (a,b).
