. This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type \Psi 1;fl with 2 IR and 0 fl 1 in the weighted function spaces B s p;q (IR n ; w(x)) and F s p;q (IR n ; w(x)) treated in [17]. Furthermore we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b2 (x)b(x; D)b1(x), where b1 (x) and b2(x) are appropriate functions and b(x; D) 2 \Psi 1;fl . Finally, on the basis of the Birman-Schwinger principle, we deal with the "negative spectrum" (bound states) of related symmetric operators in L2 . Math. Subject Classification: 46E35, 47G30, 35S05 1. Introduction The spaces B s p;q and F s p;q with s 2 IR; 0 ! p 1 (p ! 1 for the F - spaces) and 0 ! q 1 on IR n cover many well-known classical spaces such as (fractional) Sobolev spaces, H older-Zygmund spaces, Besov spaces and (inhomogeneous) Hardy spaces. In [1...

In this survey we summarise recent results concerning compact embeddings of some weighted function spaces of type B s p;q and F s p;q on R n . We study the asymptotic behaviour of their entropy and approximation numbers. We give some motivation how to apply results of this type in spectral theory. This survey provides a unified approach of recent outcomes (which have mostly been published before) and marks some present `state of the art' in this field of research. AMS Subject Classification: 46 E 35, 41 A 46, 47 B 06 Key words: (weighted) function spaces, compact embeddings, entropy numbers, approximation numbers Introduction This survey reflects some recent developments concerning the obviously symbiotic relationship between weighted function spaces on R n , entropy and approximation numbers of compact embeddings and applications to Received December 6, 1997 Analele Universitat¸ii din Craiova, Seria Matematica - Informatica, XXIV, 1997 2 dorothee d. haroske spectral theo...

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