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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**Extra resources for Notas de Matemática (49): Spectral Theory and Complex Analysis**

**Sample text**

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.