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By Paley R.E.A.C., Wiener N.

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P r o o f . Inasmuch as lim S(t) - I - 0, t40 in the norm topology of L(X), there exists (f > 0 such that IIS(t) - IIlr < 1 for each t C (0, 17]. Thus, for each t E (0, 5 ], S(t) is invertible. Let t > 5. Then there exist n E N* and 77 E [0, 5) such that t - n5 + r/. Therefore S(t) - S(5)ns(~7), and so S(t) is invertible. 3. A family of operators {G(t); t E I~} in L ( X ) is called a group of linear operators on X if(i) C(O) - I (ii) G(t § s) - G(t)G(s) for each t, s C ~. If, in addition, lira G(t) - I, t-~0 in the norm topology of L ( X ) , the group is called uniformly continuous.

Let (D,A) be a linear operator. The sets D - D(A), A(D) - R(A), {(x,y) e X x X; x E D, y - Ax} - g r a p h ( A ) a n d {x C D; Ax - 0} - ker(A) are called" the domain, the range, the graph and respectively the kernel of the operator (D, A). In the sequel, from traditional reasons, we shall denote an operator (D, A) by A " D(A) C X --+ X and we shall say that D(A) is its domain, or that A is defined on D(A), R(A) is its image and graph (A) is its graph, instead of the domain of (D, A), the image of (D, A) and respectively the graph of (D, d).

1. If {S(t) ; t _> 0} is a Co-semigroup, then the mapping (t,x) ~ S(t)x is jointly continuous from [0, +oc) x X to X. P r o o f . Let x, y E X, t _> 0 and h C R* with t + h _> 0. We distinguish between two cases" h > 0, or h < 0. 1, we deduce S(t)x. IIS(t + h ) y - S ( t ) x l l - IlS(t + h ) y - S(t + h)S(-h)x[[ _< IIS(t + h)ll~(x)lly- S(-h)xll 43 Co-semigroups. General Properties <- Me(t+h)~ (IIY -- xll + ] l S ( - h ) x - xi]), which implies that lim S ( 7 ) y - S(t)x. (~,y)-~(t-0,~) The proof is complete.

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Characters of Abelian Groups by Paley R.E.A.C., Wiener N.


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