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**Extra resources for Aspects of Positivity in Functional Analysis: Proceedings of the Conference Held on the Occasion of H.H. Schaefer's 60th Birthday, Tubingen, 24-28 …**

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N. I f u E X, let x = T It follows t h a t [Tn+lu,u] 2 n-1 = [T (T u),u] = [Tnu,Tu]. = [T(T"-'u),Tu] Therefore T clude t h a t belongs t o Class (B,). T 4 . 4 Lemma. Suppose t h a t T I lTxl 1' (a) [T2x,x] = (b) [ \TxlI2 5 (c) T2 (d) t h e spectrum Proof. Since n E N i s a r b i t r a r y , w e con- belongs t o Class (B). belongs t o Class (B1). 0 for a l l x E X; for a l l 11T2x11 1IxlI i s p o s i t i v e Hermitian; a(T) of T Then: x E X; is real. Statement ( a ) i s immediate; (b) follows from (a) by t h e Schwarz i n e q u a l i t y f o r t h e semi-inner-product.

The dimension assumption p'3, made for the treatment of balayage + theory, is replaced by the corresponding assumption that Jx separates the points of X linearly. Such harmonic spaces are called s t r o n g h a r m o n i c s p a c e s (or p-harmonic spaces with a countable base in the somewhat more general theory of CONSTANTINESCU -CORNEA C13]). Consequently, X should be from now on a strong harmonic space. p leads to such a space if and only if p > 3. The heat equation bu ==in axp+7 34 H. Bauer RP+l yields for every p 2 1 a strong harmonic space.

In fact, the following theorem of BLIEDTMER-HANSEN [8] permits a new approach to the simpliciality of S (U): _-3 . 3 Theorem. I_-. For e v e r y compact s e t KcChS(U) (fi; there exists a c o n t i n u o u s k e r n e l DK on 3 s u c h t h a t CK s o l v e s t h e " u e a k D i r i c h Z e t probZem" i n t h e f o Z Z o w i n g s e n s e : DKf EH(U1 f o r a l l f EC:Kj; f u r t h e r m o r e , DKf (XI = f (x) f o r a l l x E K. (In addition to that DK can be constructed in such a way that all measures DK (x,.