By Bernard Aupetit (auth.), Paul M. Gauthier, Gert Sabidussi (eds.)

**`**The complaints are advised if you have an interest in advanced functionality conception, strength idea, interpolation and approximation thought and similar domains.**'****Acta Sci. Mathematica (1995)**

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Ransford [Ra3] proved the following result. ) is connected for all >.. E C. ) covers all the plane except perhaps one point. P. Williams on numerical range. 15 Let a,b be two non-commuting elements of a Banach algebm A. We define W(x) = {f(x): f E A', II f 11= f(l) = I} to be the numerical mnge of x. Then U'\EC W(e'\bae-'\b) = C. ) = C. 4 The selection problem for analytic multifunctions Let K be an analytic multifunction defined on a domain D of C. 0). Unfortunately this question is very far from being solved.

By inessential perturbations, some holes may appear. For instance on 34 H B. if n 1 en+I. If n '" -1 b _ en - {eo,0, we have b of rank one, and so in JC(H), and we have D(a) Sp(a + b) = {z: Izl = I}. F. Ruston [Ru] proved that ifT E 8(X) has an essential spectral radius equal to zero, then the spectrum of T is either finite or a sequence converging to zero, and the projections associated with the non-zero spectral values have finite rank. 15, the condition that r(T) = 0 in the Calkin algebra 8(X)/JC(X) is equivalent to saying that for every f > 0 there exists an integer N such that for every n ~ N there exists Tn of finite rank with IITn - Tn II < fn.

4 The selection problem for analytic multifunctions Let K be an analytic multifunction defined on a domain D of C. 0). Unfortunately this question is very far from being solved. 11, that finite and countable analytic multifunctions have a lot of local holomorphic selections for points which are not branching points. For analytic multifunctions with zero capacity values the situation is very bad. 3, we shall give an example of this kind having no local continuous selections at all. Analytic multifunctions and their applications 21 All of this suggests very general questions: 1.