In this case, it follows from the
closed graph theorem that [M.sub.[phi]] in fact defines a bounded operator B([l.sup.2](G)) [right arrow] B([l.sup.2](G)), and the Schur norm [[parallel][phi][parallel].sub.S] of [phi] is defined to be the operator norm of [M.sub.[phi]].
The closed graph theorem shows that if B is p-admissible control operator for [(T(t)).sub.t[greater than or equal to]0] then, for some (and hence all) t [greater than or equal to] 0 there is K := [K.sub.t] [greater than or equal t] 0 such that
Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it is enough, by the closed graph theorem, to show that Range(B) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Assume that Range ([B.sub.[lambda]]) [subset or equal to] [F.sub.A] then by the closed graph theorem we obtain [B.sub.[lambda]] [member of] L (U, [F.sub.A]) which implies [B.sup.[lambda]] := ([lambda]I - [A.sub.-1])[B.sub.[lambda]] [member of] L (U,F[A.sub.-1]).
In fact, n1 clearly is a projection and n1 : A([0,2[pi]]) [right arrow] A([0,2[pi]]) is continuous by the closed graph theorem and the continuity of n (and analyticity).
The continuity of D and the closed graph theorem for webbed spaces (see [14, 24.31]) imply that [D.sub.1] is continuous (use analyticity again and notice that [A.sub.expr-] (] -[infinity], d[) and [A.sub.exp+] (]c, [infinity][) are (PLS)-spaces hence webbed).