Notas de Matemática (49): Spectral Theory and Complex by Leopoldo Nachbin and Jean Pierre Ferrier (Eds.)

By Leopoldo Nachbin and Jean Pierre Ferrier (Eds.)

Ferrier J.P. Spectral conception and complicated research (NHMS, NH, 1973)(ISBN 0444104291)

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W e can find a bounded set B in A such that (a-s)-’ exists and E belongs to B for e v e r y s off S. If s is on the boundary of S , this is the l i m i t of a sequence (s ) of the complement of S. We have P . and i f E is a Banach s p a c e of the definition of A s u ch that €3 and B B are bounded i n E, obviously ( a - s )-I is a Cauchy sequence in E . Th er ef o r e ( a - s )-I h as a P P limit x i n E such that ( a - s ) x = 1 in A, and a - s is invertible. Moreover, when s SPECTRALTHEORY 26 ranges over the boundary of S, it i s clear that (a-s)-' remains in a bounded subset of E .

T h e r e f o r e h that for e v e r y s p e c t r a l function 6 , the s e t and and there- IS s p e c t r a l A(a, , . . , an) h a s {8>0) is s p e c t r a l . Conversely, i f S is a s p e c t r a l set, the c h a r a c t e r i s t i c function Thcreforc 8 2, 'pB Ys is s p e c t r a l . * S = Min(XS,so) is a l s o a s p e c t r a l function. 4. - The holomorphic functional calculus .. We consider a b-algebra A and elements a,, , , an i n A. + xnyn = ( x , y > , when xn) and y = ( y , , , , y,) belong t o A".

I t follows f r o m Proposition 5 of Chapter II that 1 belongs to the closure of B1 i n Fl Then i f C is the unit ball of F , . 1 In other words E + ( a l - s l ) C +. + B, 1 E ( a l - s , ) ( B u C ) +. (an-sn)C. + ( a n - s n ) ( B u C ) and the statement is proved a s B u C is independent of s . Proposition 5 . - The interior of every spectral set for a l , al, . . ,a n . Proof. Let S E < ( a l , .. ,a n is spectral f o r ... , an) and choose coefficients u,(s) satisfying -sn)u ( s ) = 1 and contained i n an absolutely convex bounded set B.

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