By S. Kantorovitz

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2 Lemma. (I) For each bounded operator in ~ > O, ¢ E C÷ , I < p < =, LP(o,N) and in C[O,N]. and 0 < N < =, The same is true in J E is a LP(o,=) for c > O. (2) For ~ > O, I < p < =, 11J~l lp,N -< where C P cp ~,,I,~1/2 and 0 < N < =, ~-c,,~x{ lcl , ~} (c = c+ ~,~) depends o n l y on p and is bounded as The same e s t i m a t e p -~ oo. (,with an a d e q u a t e c o n s t a n t C) is v a ] i d f o r J ~ acting C in C[O,NI, N < =. Proof. ~) ( ; E C+, E > 0). e. on (O,N), set on (O,N) x ~ (O,N) set (x E [O,N]).

1)), IRe¢l < k Cn and 48 n+k C [a,6]-operational and t he (I) For calculus 1'_{(f) = V ( ¢ - k ) T _ k ( f ) V ( k - ~ ) , 0 < Re~ < k, D = RV({) T for T is given by f E cn+k[a,B]. is i n v a r i a n t under T0(f) and tile expression (I) simplifies to (2) z~](f) : V(-~:)To(f)V({), Proof. For IRe~il <__k, T k(f)V(k-~;) f E cn+k[a,13]. 10. and (I) is valid for all f E cn+k[a,B]. < Re¢ < k. ~ : RV(~;) is i n v a r i a n t under remain v a l i d T0(f) ~ = k %_k(f) V(k) = V ( k ) T o ( f ) . Thus, by (I) with ~ replaced by -~ (0 < Re~ < k), m T~(f) = V(-~-k)T k(f) V(k)V(~ = V(-~-k) V(k) T0(f) V(~ ) : V(-~) T 0 ( f ) V ( ~ ) , f E cn+k[~B].

V J (~;s)J+l (I) If bL(¢;~ ) (bR(~;~)) converges in Then ¢I - (s+~v). A, it is a left (right) inverse for In particular, if both converge, then R(¢;s+~v) = bL(¢;~ ) : bR(¢;~ ) . llv-nll)I/n}. 1) is valid for all 35 with both series absolutely and uniformly convergent in every compact subset of c'-~"(s). (3) For each f E H(cI*(s)) and f(s+~v) a EC, = j=O ~ (~j ) f ( j ) (s) vj = ~. ( - I ) j ( j-a ) vjf(j)(s) j=O with both series converging absolutely. the Then f o r a l l k,a exponential Take f(~) e~ (s-I-~v) = = for Volterra elements~.