By S. R. Caradus

Publication by way of S. R. Caradus, W. E. Pfaffenberger

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**Sample text**

Also I -n is zero is nilpotent, we know that the spectral radius of T -I ^ (XI - T^) has a Neumann series which terminates after p terms : ( X I - T ) " ^ = E X“"^ T^ ^ l By the uniqueness of series representations, we know that ( 2) I < n < P and B^ = 0 for n > p. , B =B -n = -n (I-P) + B P -n -n (I - P) + -n = 0 + B^^^P . -n Hence order B = 0 -n p. (vi) is the operator for n > p ^ and B -P ^ 0 so that A = O is a pole of We know that the spectral projection corresponding to B_^. Hence A = O ® It now remains to prove the converse statement.

R Likewise Q. 2) 21 THEOREM For a normed algebra onto A the mapping x x* is a homeomorphism of Q Q. Proof. Since(x*)* is continuous on Q. =X for Let a where, eventually, we let xe Q, it is enough to see thatthe mapping be afixed element of b a. Q and let b e Q First observe that a* - b* = a* O ( b o b*) - (a* о a) о b* = (a* о b) O b ’ - (a* о a) о b*. When this is expanded one readily obtains a ’ - b* = (b - a) Now set h 0. b - a =h and b* - a ’= - a * (b - a) - (b - a)b* + a* (b - a)b*. Our task is to show that к 0 if The last equations yield the inequality I|k|lál|h||(l+||a-|| + Inasmuch as b' ||a4l lib'll).

Hence if X is a M O is the associated spectral projection non zero point in sp(T^) and that in B (M), we have V 2ttí -I "-2¾:/ = C ‘O ' T)“^|M]dX / (XI - T) ^dX] I = P M O' where P^ and T c X from the remainder of Sp(T) O is the spectral projection associated with X^ and T. Now since is a suitable curve separating is a Riesz operator, we know that is P^. is finite dimensional. ■ REMARK Notice that the first part of the argument depends only on the fact that T has a connected resolvent set. e. 2) res(T).