By Bernard Aupetit

This booklet grew out of lectures on spectral idea which the writer gave on the Scuola. Normale Superiore di Pisa in 1985 and on the Universite Laval in 1987. Its target is to supply a slightly fast creation to the recent concepts of subhar monic services and analytic multifunctions in spectral conception. after all there are various paths which input the big wooded area of spectral idea: we selected to stick to these of subharmonicity and several other complicated variables in general simply because they've been found just recently and aren't but a lot frequented. In our ebook seasoned pri6t6$ $pectrale$ de$ algebre$ de Banach, Berlin, 1979, we made a primary incursion, a slightly technical one, into those newly came upon parts. for the reason that that point the trees and the thorns were reduce, so the stroll is extra agreeable and we will move even extra. which will comprehend the evolution of spectral conception from its very beginnings, you need to look at the subsequent books: Jean Dieudonne, Hutory of useful AnaIY$u, Amsterdam, 1981; Antonie Frans Monna., sensible AnaIY$i$ in Hutorical Per$pective, Utrecht, 1973; and Frederic Riesz & Bela SzOkefalvi-Nagy, Le on$ d'anaIY$e fonctionnelle, Budapest, 1952. but the photograph has replaced in view that those 3 very good books have been written. Readers may possibly persuade themselves of this via evaluating the classical textbooks of Frans Rellich, Perturbation idea, big apple, 1969, and Tosio Kato, Perturbation thought for Linear Operator$, Berlin, 1966, with the current paintings.

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Equidi s tribute d if and only if for any irr educ ible characte r X of G we have 1 n lim r; X (x . I. (X ) . l - ADIC R E PR E S E NT A T IONS i r r e ducible cha ra c te r s X of G generate a den s e sub spac e of Henc e the p r opo sition follows f r ozn leznzna 1. C (X) . - Let I-' be the Haa r zn ea sur e of G � I-' ( G) = l . Then a s e quenc e ( xn ) o f eleznent s o f X is I-' - e quidi stributed if and only if for any i r r e duc ible c ha racte r X of G , X f:. : X (x. ) n�CX) n i= l 1 = 0 This follow s frozn P rop .

Unif o rm l y distributed, if iJ. n � iJ. ;> i . e . if iJ. ;;. iJ. (f) a s n � co fo r any n f E: C ( X ) . Note that this implies that iJ. is positiv e and of total mas s said co , 1. N o t e al s o that iJ. ;> iJ. ( f ) n iJ. (f) = m e a n s that n 1 lim 1:: f(x . ) n-»co n i= 1 1 LEMMA 1 - Let (c/J ) be a family of continuous functions on X with the property that their linear c ombinati ons a r e dense in C ( X ) . Sup pas e that, for all the s equenc e (p n (c/J » n>l ha s a limit . Then the sequenc e (xn ) is equidi stributed with re spect to some measure iJ.

A ) , one has p £ (a) = = sgn (a ) IT p 'Y 00 P v P (a ) P Show that c) P1 and (a) = 'Y ' -1 a1 F =p . P d ) S h ow th at Chap . I, 1. 2. p1 c o m) in c i d e s with the char acter X 1 of = { 2 } and t:n = 1 . Show that the 2 gr oups F. , Cm , Tm , Sm coinc ide with tho s e of Exe r c ise I , hence In that the exceptional set of the c or r e sponding s ys tem is empty. (2 ) L et 2. 4. K = Q , Supp ( Linear repr e s entations of Sm We r e c all fir s t s ome well known fac t s on r e pr e s entat ion s .